Introduction and geometric constructions
An engineering drawing, a type of technical drawing, is used to fully and clearly define requirements for engineered items.
Engineering drawing (the activity) produces engineering drawings (the documents). More than merely the drawing of pictures, it is also a language—a graphical language that communicates ideas and information from one mind to another.^{[1]} Most especially, it communicates all needed information from the engineer who designed a part to the workers who will make it.
Engineering drawing and artistic drawing are both types of drawing, and either may be called simply "drawing" when the context is implicit. Engineering drawing shares some traits with artistic drawing in that both create pictures. But whereas the purpose of artistic drawing is to convey emotion or artistic sensitivity in some way (subjective impressions), the purpose of engineering drawing is to convey information (objective facts). One of the corollaries that follows from this fact is that, whereas anyone can appreciate artistic drawing (even if each viewer has his own unique appreciation), engineering drawing requires some training to understand (like any language); but there is also a high degree of objective commonality in the interpretation (also like other languages). In fact, engineering drawing has evolved into a language that is more precise and unambiguous thannatural languages; in this sense it is closer to a programming language in its communication ability. Engineering drawing uses an extensive set of conventions to convey information very precisely, with very little ambiguity.
The process of producing engineering drawings, and the skill of producing those, is often referred to as technical drawing or drafting (also spelled draughting), although technical drawings are also required for disciplines that would not ordinarily be thought of as parts of engineering (such as architecture, landscaping, cabinet making, and garmentmaking).
Persons employed in the trade of producing engineering drawings were called Draftsmen, or Draughtsmen in the past . Though these terms are still in use, recently they have been supplanted by gender neutral terms like draftsperson, or more commonly, drafter.
The various fields share many common conventions of drawing, while also having some fieldspecific conventions. For example, even within metalworking, there are some processspecific conventions to be learned—casting, machining, fabricating, and assembly all have some special drawing conventions, and within fabrication there is further division, including welding, riveting, pipefitting, and erecting. Each of these trades has some details that only specialists will have memorized.
An engineering drawing is a legal document (that is, a legal instrument), because it communicates all the needed information about "what is wanted" to the people who will expend resources turning the idea into a reality. It is thus a part of a contract; the purchase order and the drawing together, as well as any ancillary documents (engineering change orders [ECOs], calledout specs), constitute the contract. Thus, if the resulting product is wrong, the worker or manufacturer are protected from liability as long as they have faithfully executed the instructions conveyed by the drawing. If those instructions were wrong, it is the fault of the engineer. Because manufacturing and construction are typically very expensive processes (involving large amounts of capital and payroll), the question of liability for errors has great legal implications as each party tries to blame the other and assign the wasted cost to the other's responsibility. This is the biggest reason why the conventions of engineering drawing have evolved over the decades toward a very precise, unambiguous state.
Engineering drawings specify requirements of a component or assembly which can be complicated. Standards provide rules for their specification and interpretation. In 2011, a new revision of ISO 8015 was published containing the Invocation Principle. This states that, "Once a portion of the ISO GPS system is invoked in a mechanical engineering product documentation, the entire ISO GPS system is invoked." It also goes on to state that marking a drawing "Tolerancing ISO 8015" is optional. The implication of this is that any drawing using ISO symbols can only be interpreted to ISO GPS rules. The only way not to invoke the ISO GPS system is to invoke a national or other standard.
Since there are only two widely standardized definitions of size, there is only one real alternative to ISO GPS, i.e. ASME Y14.5 and Y14.5M (most recently revised in 2009). Standardization also aids internationalization, because people from different countries who speak different languages can read the same engineering drawing, and interpret it the same way. To that end, drawings should be as free of notes and abbreviations as possible so that the meaning is conveyed graphically.
For centuries, until the postWorld War II era, all engineering drawing was done manually by using pencil and pen on paper or other substrate (e.g., vellum, mylar). Since the advent of computeraided design (CAD), engineering drawing has been done more and more in the electronic medium with each passing decade. Today most engineering drawing is done with CAD, but pencil and paper have not disappeared.
Some of the tools of manual drafting include pencils, pens and their ink, straightedges, Tsquares, French curves, triangles, rulers, protractors, dividers, compasses, scales, erasers, and tacks or push pins. (Slide rules used to number among the supplies, too, but nowadays even manual drafting, when it occurs, benefits from a pocket calculator or its onscreen equivalent.) And of course the tools also include drawing boards (drafting boards) or tables. The English idiom "to go back to the drawing board", which is a figurative phrase meaning to rethink something altogether, was inspired by the literal act of discovering design errors during production and returning to a drawing board to revise the engineering drawing. Drafting machines are devices that aid manual drafting by combining drawing boards, straightedges, pantographs, and other tools into one integrated drawing environment. CAD provides their virtual equivalents.
Producing drawings usually involves creating an original that is then reproduced, generating multiple copies to be distributed to the shop floor, vendors, company archives, and so on. The classic reproduction methods involved blue and white appearances (whether whiteonblue or blueonwhite), which is why engineering drawings were long called, and even today are still often called, "blueprints" or "bluelines", even though those terms are anachronistic from a literal perspective, since most copies of engineering drawings today are made by more modern methods (often inkjet or laser printing) that yield black or multicolour lines on white paper. The more generic term "print" is now in common usage in the U.S. to mean any paper copy of an engineering drawing. In the case of CAD drawings, the original is the CAD file, and the printouts of that file are the "prints".
For centuries, engineering drawing was the sole method of transferring information from design into manufacture. In recent decades another method has arisen, called modelbased definition (MBD) or digital product definition (DPD). In MBD, the information captured by the CAD software app is fed automatically into a CAM app (computeraided manufacturing), and is translated via postprocessor into other languages such as Gcode, which is executed by a CNC machine tool (computer numerical control). Thus today it is often the case that the information travels from the mind of the designer into the manufactured component without having ever been codified by an engineering drawing. In MBD, the dataset, not a drawing, is the legal instrument. The term "technical data package" (TDP) is now used to refer to the complete package of information (in one medium or another) that communicates information from design to production (such as 3Dmodel datasets, engineering drawings, engineering change orders (ECOs), spec revisions and addenda, and so on). However, even in the MBD era, where theoretically production could happen without any drawings or humans at all, it is still the case that drawings and humans are involved. It still takes CAD/CAM programmers, CNC setup workers, and CNC operators to do manufacturing, as well as other people such as quality assurance staff (inspectors) and logistics staff (for materials handling, shippingandreceiving, and front office functions). These workers often use drawings in the course of their work that have been produced by rendering and plotting (printing) from the MBD dataset. When proper procedures are being followed, a clear chain of precedence is always documented, such that when a person looks at a drawing, s/he is told by a note thereon that this drawing is not the governing instrument (because the MBD dataset is). In these cases, the drawing is still a useful document, although legally it is classified as "for reference only", meaning that if any controversies or discrepancies arise, it is the MBD dataset, not the drawing, that governs.
Almost all engineering drawings (except perhaps referenceonly views or initial sketches) communicate not only geometry (shape and location) but also dimensions andtolerances for those characteristics. Several systems of dimensioning and tolerancing have evolved. The simplest dimensioning system just specifies distances between points (such as an object's length or width, or hole center locations). Since the advent of welldeveloped interchangeable manufacture, these distances have been accompanied by tolerances of the plusorminus or minandmaxlimit types. Coordinate dimensioning involves defining all points, lines, planes, and profiles in terms of Cartesian coordinates, with a common origin. Coordinate dimensioning was the sole best option until the postWorld War II era saw the development of geometric dimensioning and tolerancing (GD&T), which departs from the limitations of coordinate dimensioning (e.g., rectangularonly tolerance zones, tolerance stacking) to allow the most logical tolerancing of both geometry and dimensions (that is, both form [shapes/locations] and sizes).
Drawings convey the following critical information:
A variety of line styles graphically represent physical objects. Types of lines include the following:
Lines can also be classified by a letter classification in which each line is given a letter.
In most cases, a single view is not sufficient to show all necessary features, and several views are used. Types of views include the following:
The orthographic projection shows the object as it looks from the front, right, left, top, bottom, or back, and are typically positioned relative to each other according to the rules of either firstangle or thirdangle projection. The origin and vector direction of the projectors (also called projection lines) differs, as explained below.
Until the late 19th century, firstangle projection was the norm in North America as well as Europe; but circa the 1890s, the meme of thirdangle projection spread throughout the North American engineering and manufacturing communities to the point of becoming a widely followed convention, and it was an ASA standard by the 1950s. Circa World War I, British practice was frequently mixing the use of both projection methods.
As shown above, the determination of what surface constitutes the front, back, top, and bottom varies depending on the projection method used.
Not all views are necessarily used. Generally only as many views are used as are necessary to convey all needed information clearly and economically.The front, top, and rightside views are commonly considered the core group of views included by default, but any combination of views may be used depending on the needs of the particular design. In addition to the 6 principal views (front, back, top, bottom, right side, left side), any auxiliary views or sections may be included as serve the purposes of part definition and its communication. View lines or section lines (lines with arrows marked "AA", "BB", etc.) define the direction and location of viewing or sectioning. Sometimes a note tells the reader in which zone(s) of the drawing to find the view or section.
An auxiliary view is an orthographic view that is projected into any plane other than one of the six principal views. These views are typically used when an object contains some sort of inclined plane. Using the auxiliary view allows for that inclined plane (and any other significant features) to be projected in their true size and shape. The true size and shape of any feature in an engineering drawing can only be known when the Line of Sight (LOS) is perpendicular to the plane being referenced. It is shown like a threedimensional object.
The isometric projection show the object from angles in which the scales along each axis of the object are equal. Isometric projection corresponds to rotation of the object by ± 45° about the vertical axis, followed by rotation of approximately ± 35.264° [= arcsin(tan(30°))] about the horizontal axis starting from an orthographic projection view. "Isometric" comes from the Greek for "same measure". One of the things that makes isometric drawings so attractive is the ease with which 60 degree angles can be constructed with only acompass and straightedge.
Isometric projection is a type of axonometric projection. The other two types of axonometric projection are:
An oblique projection is a simple type of graphical projection used for producing pictorial, twodimensional images of threedimensional objects:
In both oblique projection and orthographic projection, parallel lines of the source object produce parallel lines in the projected image.
Perspective is an approximate representation on a flat surface, of an image as it is perceived by the eye. The two most characteristic features of perspective are that objects are drawn:
Projected views (either Auxiliary or Orthographic) which show a cross section of the source object along the specified cut plane. These views are commonly used to show internal features with more clarity than may be available using regular projections or hidden lines. In assembly drawings, hardware components (e.g. nuts, screws, washers) are typically not sectioned.
Plans are usually "scale drawings", meaning that the plans are drawn at specific ratio relative to the actual size of the place or object. Various scales may be used for different drawings in a set. For example, a floor plan may be drawn at 1:50 (1:48 or 1/4"=1'0") whereas a detailed view may be drawn at 1:25 (1:24 or 1/2"=1'0"). Site plans are often drawn at 1:200 or 1:100.
Scale is a nuanced subject in the use of engineering drawings. On one hand, it is a general principle of engineering drawings that they are projected using standardized, mathematically certain projection methods and rules. Thus, great effort is put into having an engineering drawing accurately depict size, shape, form, aspect ratios between features, and so on. And yet, on the other hand, there is another general principle of engineering drawing that nearly diametrically opposes all this effort and intent—that is, the principle that users are not to scale the drawing to infer a dimension not labeled. This stern admonition is often repeated on drawings, via a boilerplate note in the title block telling the user, "DO NOT SCALE DRAWING."
The explanation for why these two nearly opposite principles can coexist is as follows. The first principle—that drawings will be made so carefully and accurately—serves the prime goal of why engineering drawing even exists, which is successfully communicating part definition and acceptance criteria—including "what the part should look like if you've made it correctly." The service of this goal is what creates a drawing that one even could scale and get an accurate dimension thereby. And thus the great temptation to do so, when a dimension is wanted but was not labeled. The second principle—that even though scaling the drawing will usually work, one should nevertheless never do it—serves several goals, such as enforcing total clarity regarding who has authority to discern design intent, and preventing erroneous scaling of a drawing that was never drawn to scale to begin with (which is typically labeled "drawing not to scale" or "scale: NTS"). When a user is forbidden from scaling the drawing, s/he must turn instead to the engineer (for the answers that the scaling would seek), and s/he will never erroneously scale something that is inherently unable to be accurately scaled.
But in some ways, the advent of the CAD and MBD era challenges these assumptions that were formed many decades ago. When part definition is defined mathematically via a solid model, the assertion that one cannot interrogate the model—the direct analog of "scaling the drawing"—becomes ridiculous; because when part definition is defined this way, it is not possible for a drawing or model to be "not to scale". A 2D pencil drawing can be inaccurately foreshortened and skewed (and thus not to scale), yet still be a completely valid part definition as long as the labeled dimensions are the only dimensions used, and no scaling of the drawing by the user occurs. This is because what the drawing and labels convey is in reality a symbol of what is wanted, rather than a true replica of it. (For example, a sketch of a hole that is clearly not round still accurately defines the part as having a true round hole, as long as the label says "10mm DIA", because the "DIA" implicitly but objectively tells the user that the skewed drawn circle is a symbolrepresenting a perfect circle.) But if a mathematical model—essentially, a vector graphic—is declared to be the official definition of the part, then any amount of "scaling the drawing" can make sense; there may still be an error in the model, in the sense that what was intended is not depicted (modeled); but there can be no error of the "not to scale" type—because the mathematical vectors and curves are replicas, not symbols, of the part features.
Even in dealing with 2D drawings, the manufacturing world has changed since the days when people paid attention to the scale ratio claimed on the print, or counted on its accuracy. In the past, prints were plotted on a plotter to exact scale ratios, and the user could know that a line on the drawing 15mm long corresponded to a 30mm part dimension because the drawing said "1:2" in the "scale" box of the title block. Today, in the era of ubiquitous desktop printing, where original drawings or scaled prints are often scanned on a scanner and saved as a PDF file, which is then printed at any percent magnification that the user deems handy (such as "fit to paper size"), users have pretty much given up caring what scale ratio is claimed in the "scale" box of the title block. Which, under the rule of "do not scale drawing", never really did that much for them anyway.
The required sizes of features are conveyed through use of dimensions. Distances may be indicated with either of two standardized forms of dimension: linear and ordinate.
Sizes of circular features are indicated using either diametral or radial dimensions. Radial dimensions use an "R" followed by the value for the radius; Diametral dimensions use a circle with forwardleaning diagonal line through it, called the diameter symbol, followed by the value for the diameter. A radiallyaligned line with arrowhead pointing to the circular feature, called a leader, is used in conjunction with both diametral and radial dimensions. All types of dimensions are typically composed of two parts: the nominal value, which is the "ideal" size of the feature, and the tolerance, which specifies the amount that the value may vary above and below the nominal.
Sizes of drawings typically comply with either of two different standards, ISO (World Standard) or ANSI/ASME Y14 (American), according to the following tables:
A4  210 X 297 

A3  297 X 420 
A2  420 X 594 
A1  594 X 841 
A0  841 X 1189 
A  8.5" X 11" 

B  11" X 17" 
C  17" X 22" 
D  22" X 34" 
E  34" X 44" 
D1  24" X 36" 

E1  30" X 42" 
H  larger still [intracompany standards] 
I  larger still [intracompany standards] 
J  larger still [intracompany standards] 
The metric drawing sizes correspond to international paper sizes. These developed further refinements in the second half of the twentieth century, when photocopying became cheap. Engineering drawings could be readily doubled (or halved) in size and put on the next larger (or, respectively, smaller) size of paper with no waste of space. And the metric technical pens were chosen in sizes so that one could add detail or drafting changes with a pen width changing by approximately a factor of the square root of 2. A full set of pens would have the following nib sizes: 0.13, 0.18, 0.25, 0.35, 0.5, 0.7, 1.0, 1.5, and 2.0 mm. However, the International Organization for Standardization (ISO) called for four pen widths and set a colour code for each: 0.25 (white), 0.35 (yellow), 0.5 (brown), 0.7 (blue); these nibs produced lines that related to various text character heights and the ISO paper sizes.
All ISO paper sizes have the same aspect ratio, one to the square root of 2, meaning that a document designed for any given size can be enlarged or reduced to any other size and will fit perfectly. Given this ease of changing sizes, it is of course common to copy or print a given document on different sizes of paper, especially within a series, e.g. a drawing on A3 may be enlarged to A2 or reduced to A4.
The U.S. customary "Asize" corresponds to "letter" size, and "Bsize" corresponds to "ledger" or "tabloid" size. There were also once British paper sizes, which went by names rather than alphanumeric designations.
American Society of Mechanical Engineers (ASME) Y14.2, Y14.3, and Y14.5 are commonly referenced standards in the U.S.
Technical lettering is the process of forming letters, numerals, and other characters in technical drawing. It is used to describe, or provide detailed specifications for, an object. With the goals of legibility and uniformity, styles are standardized and lettering ability has little relationship to normal writing ability. Engineering drawings use a Gothic sansserifscript, formed by a series of short strokes. Lower case letters are rare in most drawings of machines. ISO Lettering templates, designed for use with technical pens and pencils, and to suit ISO paper sizes, produce lettering characters to an international standard. The stroke thickness is related to the character height (for example, 2.5mm high characters would have a stroke thickness  pen nib size  of 0.25mm, 3.5 would use a 0.35mm pen and so forth). The ISO character set (font) has a seriffed one, a barred seven, an open four, six, and nine, and a round topped three, that improves legibility when, for example, an A0 drawing has been reduced to A1 or even A3 (and perhaps enlarged back or reproduced/faxed/ microfilmed &c). When CAD drawings became more popular, especially using US American software, such as AutoCAD, the nearest font to this ISO standard font was Romantic Simplex (RomanS)  a proprietary shx font) with a manually adjusted width factor (over ride) to make it look as near to the ISO lettering for the drawing board. However, with the closed four, and arced six and nine, romans.shx typeface could be difficult to read in reductions. In more recent revisions of software packages, the TrueTypefont ISOCPEUR reliably reproduces the original drawing board lettering stencil style, however, many drawings have switched to the ubiquitous Arial.ttf.
The title block (T/B, TB) is an area of the drawing that conveys headertype information about the drawing, such as:
Traditional locations for the title block are the bottom right (most commonly) or the top right or center.
The revisions block (rev block) is a tabulated list of the revisions (versions) of the drawing, documenting the revision control.
Traditional locations for the revisions block are the top right (most commonly) or adjoining the title block in some way.
The next assembly block, often also referred to as "where used" or sometimes "effectivity block", is a list of higher assemblies where the product on the current drawing is used. This block is commonly found adjacent to the title block.
The notes list provides notes to the user of the drawing, conveying any information that the callouts within the field of the drawing did not. It may include general notes, flagnotes, or a mixture of both.
Traditional locations for the notes list are anywhere along the edges of the field of the drawing.
General notes (G/N, GN) apply generally to the contents of the drawing, as opposed to applying only to certain part numbers or certain surfaces or features.
Flagnotes or flag notes (FL, F/N) are notes that apply only where a flagged callout points, such as to particular surfaces, features, or part numbers. Typically the callout includes a flag icon. Some companies call such notes "delta notes", and the note number is enclosed inside a triangular symbol (similar to capital letter delta, Δ). "FL5" (flagnote 5) and "D5" (delta note 5) are typical ways to abbreviate in ASCIIonly contexts.
The field of the drawing (F/D, FD) is the main body or main area of the drawing, excluding the title block, rev block, and so on.
The list of materials (L/M, LM, LoM), bill of materials (B/M, BM, BoM), or parts list (P/L, PL) is a (usually tabular) list of the materials used to make a part, and/or the parts used to make an assembly. It may contain instructions for heat treatment, finishing, and other processes, for each part number. Sometimes such LoMs or PLs are separate documents from the drawing itself.
Traditional locations for the LoM/BoM are above the title block, or in a separate document.
Some drawings call out dimensions with parameter names (that is, variables, such a "A", "B", "C"), then tabulate rows of parameter values for each part number.
Traditional locations for parameter tables, when such tables are used, are floating near the edges of the field of the drawing, either near the title block or elsewhere along the edges of the field.
Each view or section is a separate set of projections, occupying a contiguous portion of the field of the drawing. Usually views and sections are called out with crossreferences to specific zones of the field.
Often a drawing is divided into zones by a grid, with zone labels along the margins, such as A,B,C,D up the sides and 1,2,3,4,5,6 along the top and bottom. Names of zones are thus, for example, A5, D2, or B1. This feature greatly eases discussion of, and reference to, particular areas of the drawing.
As in many technical fields, a wide array of abbreviations and symbols have been developed in engineering drawing during the 20th and 21st centuries. For example, cold rolled steel is often abbreviated as CRS, and diameter is often abbreviated as DIA, D, or ⌀.
Here is an example of an engineering drawing (an isometric view of the same object is shown above). The different line types are colored for clarity.
Graphical projection 

Planar[show]

Other[show]

Views[show]

Topics[show]


Graphical projection is a protocol by which an image of a threedimensional object is projected onto a planar surface without the aid of numerical calculation, used in technical drawing.^{[}
The projection is achieved by the use of imaginary "projectors". The projected, mental image becomes the technician’s vision of the desired, finished picture. By following the protocol the technician may produce the envisioned picture on a planar surface such as drawing paper. The protocols provide a uniform imaging procedure among people trained in technical graphics (mechanical drawing, computer aided design, etc.).
There are two graphical projection categories each with its own protocol:
Isometric projection.
Oblique projection.
Oblique projection.
Onepoint perspective projection.
In parallel projection,the lines of sight from the object to the projection plane are parallel to each other. Within parallel projection there is an ancillary category known as "pictorials". Pictorials show an image of an object as viewed from a skew direction in order to reveal all three directions (axes) of space in one picture. Because pictorial projections innately contain this distortion, in the rote, drawing instrument for pictorials, some liberties may be taken for economy of effort and best effect.it is a simultaneous process of viewing the image give pictures
The Orthographic projection is derived from the principles of descriptive geometry and is a twodimensional representation of a threedimensional object. It is a parallel projection (the lines of projection are parallel both in reality and in the projection plane). It is the projection type of choice for working drawings.
Within parallel projection there is a subcategory known as Pictorials. Pictorials show an image of an object as viewed from a skew direction in order to reveal all three directions (axes) of space in one picture. Parallel projection pictorial instrument drawings are often used to approximate graphical perspective projections, but there is attendant distortion in the approximation. Because pictorial projections inherently have this distortion, in the instrument drawing of pictorials, great liberties may then be taken for economy of effort and best effect. Parallel projection pictorials rely on the technique of axonometric projection ("to measure along axes").
Axonometric projection is a type of parallel projection used to create a pictorial drawing of an object, where the object is rotated along one or more of its axes relative to the plane of projection.^{[1]}
There are three main types of axonometric projection: isometric, dimetric, and trimetric projection.
In isometric pictorials (for protocols see isometric projection), the direction of viewing is such that the three axes of space appear equally foreshortened, and there is a common angle of 60° between them. As the distortion caused by foreshortening is uniform the proportionality of all sides and lengths are preserved, and the axes share a common scale. This enables measurements to be read or taken directly from the drawing.
In dimetric pictorials (for protocols see dimetric projection), the direction of viewing is such that two of the three axes of space appear equally foreshortened, of which the attendant scale and angles of presentation are determined according to the angle of viewing; the scale of the third direction (vertical) is determined separately. Approximations are common in dimetric drawings.
In trimetric pictorials (for protocols see trimetric projection), the direction of viewing is such that all of the three axes of space appear unequally foreshortened. The scale along each of the three axes and the angles among them are determined separately as dictated by the angle of viewing. Approximations in Trimetric drawings are common.
In oblique projections the parallel projection rays are not perpendicular to the viewing plane as with orthographic projection, but strike the projection plane at an angle other than ninety degrees. In both orthographic and oblique projection, parallel lines in space appear parallel on the projected image. Because of its simplicity, oblique projection is used exclusively for pictorial purposes rather than for formal, working drawings. In an oblique pictorial drawing, the displayed angles among the axes as well as the foreshortening factors (scale) are arbitrary. The distortion created thereby is usually attenuated by aligning one plane of the imaged object to be parallel with the plane of projection thereby creating a true shape, fullsize image of the chosen plane. Special types of oblique projections are:
In cavalier projection (sometimes cavalier perspective or high view point) a point of the object is represented by three coordinates, x, y and z. On the drawing, it is represented by only two coordinates, x" and y". On the flat drawing, two axes, x and z on the figure, are perpendicular and the length on these axes are drawn with a 1:1 scale; it is thus similar to the dimetric projections, although it is not an orthographic projection, as the third axis, here y, is drawn in diagonal, making an arbitrary angle with the x" axis, usually 30 or 45°. The length of the third axis is not scaled.
The term cabinet projection (sometimes cabinet perspective) stems from its use in illustrations by the furniture industry.^{[citation needed]} Like cavalier perspective, one face of the projected object is parallel to the viewing plane, and the third axis is projected as going off in an angle (typical 30° or 45° or arctan(2)=63.4°). Unlike cavalier projection, where the third axis keeps its length, with cabinet projection the length of the receding lines is cut in half.
Perspective projection is a linear projection where three dimensional objects are projected on a picture plane. This has the effect that distant objects appear smaller than nearer objects.
It also means that lines which are parallel in nature (that is, meet at the point at infinity) appear to intersect in the projected image, for example if railways are pictured with perspective projection, they appear to converge towards a single point, called vanishing point. Photographic lenses and the human eye work in the same way, therefore perspective projection looks most realistic.^{[2]} Perspective projection is usually categorized into onepoint, twopoint and threepoint perspective, depending on the orientation of the projection plane towards the axes of the depicted object.^{[3]}
Graphical projection methods rely on the duality between lines and points, whereby two straight lines determine a point while two points determine a straight line. The orthogonal projection of the eye point onto the picture plane is called the principal vanishing point (P.P. in the scheme on the left, from the Italian term punto principale, coined during the renaissance).^{[4]}
Two relevant points of a line are:
The principal vanishing point is the vanishing point of all horizontal lines perpendicular to the picture plane. The vanishing points of all horizontal lines lie on the horizon line, obviously. If, as is often the case, the picture plane is vertical, all vertical lines are drawn vertically, and have no finite vanishing point on the picture plane. Various graphical methods can be easily envisaged for projecting geometrical scenes. For example, lines traced from the eye point at 45° to the picture plane intersect the latter along a circle whose radius is the distance of the eye point from the plane, thus tracing that circle aids the construction of all the vanishing points of 45° lines; in particular, the intersection of that circle with the horizon line consists of two distance points. They are useful for drawing chessboard floors which, in turn, serve for locating the base of objects on the scene. In the perspective of a geometric solid on the right, after choosing the principal vanishing point —which determines the horizon line— the 45° vanishing point on the left side of the drawing completes the characterization of the (equally distant) point of view. Two lines are drawn from the orthogonal projection of each vertex, one at 45° and one at 90° to the picture plane. After intersecting the ground line, those lines go toward the distance point (for 45°) or the principal point (for 90°). Their new intersection locates the projection of the map. Natural heights are measured above the ground line and then projected in the same way until they meet the vertical from the map.
Drawing layout and Lettering
Layout of a drawing
sheet
Every drawing sheet is to follow a
particular layout. As a standard practice sufficient margins
are to be provided on all sides of the drawingsheet. The
drawing sheet should have drawing space and title page. A
typical layout of a drawing sheet is shown in the figure
below:
Figure 1. A typical layout of a drawing sheet.
Layout of the title box
recommended for Engineering Drawing Course
The title box shown in figure 2 can be used for
the engineering Drawing Course.
Figure 2. A typical title box recommended for Engineering students.
Lettering
Lettering is used for writing of
titles, subtitles, dimensions, scales and other details on a
drawing. Typical lettering features used for engineering
drawing is shown in figure 3. The following rules are to be
followed in lettering. The letter sizes generally recommended
for various items are shown in Table 1.
The Indian standard followed for lettering is BIS: 9609
Lettering types generally used for creating a drawing are
Table 2 and Table 3 indicates the specifications for Type A and Type B letters.
Figure 3. Typical lettering features.
Heights of Letters and Numerals
Table 1 The letter sizes recommended for various items
Table 2. Specifications of A Type Lettering
Table 3. Specifications of B Type Lettering
How
to begin your drawing?
To
start with the preparation of a drawing the procedure mentioned
below may be followed:
Keeping the drawing clean is a must
Lines and Dimensioning
Lines
Lines is one important aspect of
technical drawing. Lines are always used to construct
meaningful drawings. Various types of lines are used to construct
drawing, each line used in some specific sense. Lines are
drawn following standard conventions mentioned in BIS (SP46:2003).
A line may be curved, straight, continuous, segmented. It may be
drawn as thin or thick. A few basic types of lines widely used in
drawings are shown in Table 1.
Table 1. Types of letters used in engineering drawing.
Line Strokes
Line strokes refer to the directions
of drawing straight and curved lines. The standards for lines is
given in BIS : SP46, 2003
Vertical and inclined lines are
drawn from top to bottom, horizontal lines are drawn from left to
right. Curved lines are drawn from left to right or top to bottom.
The direction of strokes are illustrated in figure 1.
Figure 1. The line strokes for drawing straight and curved lines.
Conventions used in lines
Typical figures showing various lines used in the construction of engineering drawing is shown in figure 2.
Figure 2 Typical figure showing various lines used engineering drawing
A typical use of various lines in an engineering drawing is shown in figure below:
Dimensioning
The size and other details of the
object essential for its construction and function, using lines,
numerals, symbols, notes, etc are required to be indicated in a
drawing by proper dimensioning. These dimensions indicated should
be those that are essential for the production, inspection and
functioning of the object and should be mistaken as those that are
required to make the drawing of an object. The dimensions are
written either above the dimension lines or inserted at the middle
by breaking the dimension lines.
Normally two types of dimensioning system exist. i.e. Aligned system and the unidirectional system.These are shown in figure 3.
In the aligned system the
dimensions are placed perpendicular to the dimension line in such a
way that it may be read from bottom edge or right hand edge of the
drawing sheet. The horizontal and inclined dimension can be read
from the bottom where as all the vertical dimensions can be read
from the right hand side of the drawing sheet.
In the unidirectional system, the
dimensions are so oriented such that they can be read from the
bottom of the drawing.
Figure 3. The aligned system and unidirectional system of dimensioning.
Rules to be followed for dimensioning. Refer figure 4.
Figure 4. typical dimension lines
Dimensioning consists of the following:
The important elements of dimensioning consists of extension lines, leader line, arrows and dimensions.
Extension line – a thin,
solid line perpendicular to a dimension line, indicating which
feature is associated with the dimension. There should be a visible
gap of 1.5 mm between the feature’s corners and the end of the
extension line.Figure 5 shows extension lines.
Leader line
A thin, solid line used to indicate
the feature with which a dimension, note, or symbol is
associated. Generally this is a straight line
drawn at an angle that is neither horizontal nor vertical.
Leader line is terminated with an arrow touching the part or
detail. On the end opposite the arrow, the leader line will
have a short, horizontal shoulder. Text is extended from this
shoulder such that the text height is centered with the
shoulder line
Figure 5. showing extension lines
Figure 6.Various types of arrows used for dimensioning
The specification of dimension lines are shown in figure 7.
Figure 7 showing the specification of dimension lines.
Dimensioning of angles: The normal convention for dimensioning of angles are illustrated in figure 8.
Figure 8 conventions used for dimensioning angles.
Few examples during dimensioning of solids are shown below:
RULES OF DIMENSIONING
Geometrical Constructions: Part1
Geometric Construction
Drawing consists of construction of primitive geometric forms viz. points, lines and planes that serve a the building blocks for more complicated geometric shapes and defining the position of object in space.
The use of lines for obtaining the drawing of planes is shown in figure 1.
Figure 1 illustrates various planes generally encountered
Solids are obtained by combination of planes. Plane surfaces of simple solids are shown in figure 2.
Figure 2 surfaces of few simples solids .
In addition curved surfaces also exists. Figure 3 shows some of solids having curved surfaces.
Figure 3. Solids having curved surfaces.
Primitive geometric
forms
The shapes of objects are formed
from primitive geometric forms . These are
The basic 2D geometric primitives, from which other more complex geometric forms are derived.
Point
A point is a theoretical location that has neither width, height, nor depth and describes exact location in space. A point is represented in technical drawing as a small cross made of dashes that are approximately 3 mm long. As shown in figure 4, a point is used to mark the locations of centers and loci, the intersection ends, middle of entities
Figure 4. shows the various use of points.
Line
A line is a geometric primitive that has length and direction, but no thickness. Lines may be straight, curved or a combination of these. As shown in figure 5, lines have few important relationship or conditions, such as parallel, intersecting, and tangent. Lines can be of specific length or nonspecific length. A Ray is a sStraight line that extends to infinity from a specified point.
Figure 5. Relationship of one line to another line or arc
Bisecting a line
The procedure of bisecting a
given line AB is illustrated in figure 6.
With A as centre and radius equal to
higher than half AB, draw two arcs. With B as centre and with the
same radius draw another arc intersecting the preious arcs. The
line joining the intersection points is the perpendicular bisector
of the line AB.
Figure 6. Illustrates the method of bisecting a line
Dividing a line into equal parts
The method of dividing a line MO into equal number of parts is illustrated in figure 7.
Figure 7. Dividing a line in to equal number of parts.
Planar tangent condition exists when two geometric forms meet at a single point and do not intersect. This is self explanatory from figure 8.
Figure 8. Illustrates the existence of planar tangent condition.
Locating tangent points on circle and arcs
The method of locating tangent points on circle and arcs as well as thhe common tangent to two circles are shown in figure 9(a) and (b) .
Figure 9. Locating the tangent points to arcs or circles.
Drawing an arc tangent to a given
point on the line
The steps for drawing the arc
tangent to a given point on a line is shown in figure
10.
Figure 10. Drawing an arc tangent to the a given point on a line.
Drawing an arc, tangent to two lines
The steps used to drawn an arc tangent to two lines is illustrated in figure 11.
Figure 11. illustrates the method of drawing an arc tangent to two lines.
Drawing an arc, tangent to a line and an arc
Figure 12 shows the steps in drawing an arc tangent to a line and an arc that (a) that do not intersect and (b) that intersect each other.
Figure 12 Drawing an arc tangent to a line and an arc
Geometric construction: Part2
Construction of Regular Polygon of given length AB
To construct a regular polygon with length of edge AB us shown in figure 1.
Figure 1. Construction of a regular polygon with a given length of edge.
General method of drawing any polygon
A more general method of drawing any polygon with a given length of edge is shown in figure 2.
Figure 2 Drawing any polygon with a given length of edge
Inscribe a circle inside a regular polygon
The method of inscribing a circle inside a regular polygon is illustrated in figure 3.
Figure 3 Inscribing a circle inside a regular polygon
Inscribe a regular polygon of any number of sides (say n = 5), in a circle
Figure 4 shows the method of inscribing a regular polygon of any number of sides.
Figure 4 Inscribing a regular polygon of any number of sides.
To draw a circle to touch a given line, and a given circle at a given point on it.
The method is illustrated in figure 5.
Figure 5. shows the method of drawing a circle to touch a given line and a given circle at a particular point.
Inside a regular polygon, draw the same number of equal circles as the side of the polygon,each circle touching one side of the polygon and two of the other circles
The technique is shown in figure 6
Figure 6. Drawing the same number of equal circles, in a given polygon , as the side of the polygon
Figure 7 shows the technique for drawing the same number of equal circles as the side of the polygon inside a regular polygon, each circle touching two adjacent sides of the polygon and two of the other circles.
Figure 7. drawing the same number of equal circles as the side of the polygon inside a regular polygon
Figure 8 shows the method of drawing a circle touching three lines inclined to each other but not forming a triangle.
Figure 9 shows the method of drawing outside a regular polygon, the same number of equal circles as the sides of the polygon, each circle touching one side of the polygon and two of the other circles.
Figure 9 shows the method of drawing outside a regular polygon, the same number of equal circles as the sides of the polygon,
Scales
Scales
There is a wide variation in
sizes for engineering objects. Some are very large (eg. Aero
planes, rockets, etc) Some are vey small ( wrist watch, MEMs
components)
There is a need to reduce or enlarge
while drawing the objects on paper. Some objects can be drawn to
their actual size. The proportion by which the drawing of aan
object is enlarged or reduced is called the scale of the
drawing.
Definition
A scale is defined as the ratio of
the linear dimensions of the object as represented in a
drawing to the actual dimensions of the same.
BIS Recommended Scales are shown in table 1.
Table 1. The common scales recommended.
Intermediate scales can be used in exceptional cases where recommended scales can not be applied for functional reasons.
Types of Scale :
Engineers Scale :
The relation between the dimension on the drawing and the
actual dimension of the object is mentioned numerically (like
10 mm = 15 m).
Graphical Scale: Scale is drawn on the drawing itself. This takes care of the shrinkage of the engineer’s scale when the drawing becomes old.
Types of Graphical Scale :
Representative fraction (R.F.) :
When a 1 cm long line in a drawing represents 1 meter length of the object
Length of scale = RF x Maximum distance to be represented
Plain scale :
Construct a plain scale of RF = 1:4, to show centimeters and long enough to measure up to 5 decimeters.
Diagonal Scale :
Diagonal scale…..Concept
Construct a Diagonal scale of RF = 3:200 showing meters, decimeters and centimeters. The scale should measure up to 6 meters. Show a distance of 4.56 meters
Diagonal Scale
Vernier Scale
Least Count (LC) is
the minimum length that can be measured precisely by a given
vernier scale. This can be determined by the following
expression:
LC = MSD –
VSD ( if MSD > VSD)
LC = VSD –
MSD ( if VSD > MSD)
The LC is mentioned as a fraction of
the MSD.
If the MSD of a scale represents 1
mm and LC is 0.1 mm,
LC = 0.1 mm = (1/10) MSD
This length must be divided in to 10 equal parts so that LC = 0.1 mm
Backward Vernier scale
Question: Draw a Vernier scale of R.F. = 1/25 to read up to 4 meters. On it show lengths 2.39 m and 0.91 m
Comparative Scales
1 Mile = 8 fur. = 1760 yd = 5280 ft
Construct a plain comparative
Scales of RF = 1/624000 to read up to 50 kms and 40
miles.
On these show the kilometer
equivalent to 18 miles
Draw a 4 in. line AC and construct a
plain scale to represent mile and 8cm line AB and construct the
kilometer scale below the mile scale.
On the mile scale, determine the
distance equal to 18 miles (PQ)
Mark P’Q’ = PQ on the kilometer
scale such that P’ will coincide with the appropriate main
division. Find the length represented by
P’Q’.
P’Q’ = 29 km.
(1Mile = 1.60934
km)
Scale of chords
Scale of chords is used to
measure angles when a protractor is not available, by comparing the
angles subtended by chords of an arc at the centre of the
arc.
Draw a line AO of any suitable
length.
At O, erect a perpendicular OB such
that OB – OA
With O as centre, draw an arc
AB
Divide the arc in to 9 equal
parts by the following method.
The total length of AB is now
divided in to 9 equal parts. Number the divisions as 10, 20,
30, 40 ,etc.
Transfer all the divisions on the
arc to th line AO by drawing arcs with A as a centre and radii
equal to the chords A10, 1020, 2030, …. AB.
Construct the linear degree scale by
drawing the rectangles below AC. Mark the divisions in the
rectangle with zero below A and number the divisions subsequently
as 10^{o}, 20^{o}, 30^{o}, 40^{o},
….., 90^{o}
Engineering curves: Ellipse
Conic curves
(conics)
Curves formed by the intersection of
a plane with a right circular cone. e.g. Parabola, hyperbola and
ellipse. Right circular cone is a cone that has a circular base and
the axis is inclined at 900 to the base and passes
through the center of the base.Conic
sections are always "smooth". More precisely, they never contain
any inflection points. This is important for many
applications, such as aerodynamics, civil engineering, mechanical
engineering, etc.Figure 1. Shows a right cone and the various conic
curves that can be obtained from a cone by sectioning the cone at
various conditions.
Figure 1. Shows a right cone and the various conic curves that can be obtained from a cone by sectioning the cone at various conditions.
Conic
Conic is defined as the locus of a point moving in a plane such that the ratio of its distance from a fixed point and a fixed straight line is always constant.
This is illustrated in figure 2.
Figure 2. illustrates the directrices and foci of a conic curve.
When
eccentricity
< 1 Ellipse
=1 Parabola
> 1
Hyperbola
eg. when e=1/2, the curve is an Ellipse, when e=1, it is a parabola and when e=2, it is a hyperbola. Figure 3 shows the ellipse, parabola and hyperbola.
Figure 3 shows the relationship of eccentricity with different conic curves.
Ellipse
Referring to figure 4, an ellipse can be defined in the following ways.
Construction of Ellipse
Figure 4. illustrating an ellipse.
FocusDirectrix or Eccentricity
Method
Given : the distance of focus from
the directrix and eccentricity
Figure 5. shows the method of
drawing an ellipse if the distance of focus from the directrix is
80 mm and the eccentricity is 3/4.
Figure 5. drawing an ellipse if the distance of focus from the directrix and the eccentricity is given
5.With F as a centre and radius = 1–1’, cut two arcs on the perpendicular through 1 to locate P1 and P1’. Similarly, with F as centre and radii = 2–2’, 3–3’, etc., cut arcs on the corresponding perpendiculars to locate P2 and P2’, P3 and P3’, etc. Also, cut similar arcs on the perpendicular through O to locate V1 and V1’.
6.Draw a smooth closed curve passing through V, P1, P/2, P/3, …, V1, …, V’, …, V1’, … P/3’, P/2’, P1’.
7.Mark F’ on CC’ such that V’ F’ = VF.
An ellipse is also the set of all points in a plane for which the sum of the distances from the two fixed points (the foci) in the plane is constant.This is clear from figure 6.
Figure 6.Another definition of ellipse
Arcs of Circle
Method
The arc of circle method of drawing
an ellipse is generally used when (i) the major axies and
minor axis are known, and (ii) the major axis and the distance
between the foci are know. Themethod of drawing the
ellipse by the arcs of circle method is as follows and is shown in
figure 7.
Draw AB & CD perpendicular to
each other as the major diameter minor diameter
respectively.
With centre as C or D, and half the
major diameter as radius draw arcs to intersect the major diameter
to obtain the foci at X and Y.
Mark a number of points along line
segment XY and number them. Points need not be
equidistant.
Set the compass to radius B1
and draw two arcs, with Y as center. Set the compass to
radius A1, and draw two arcs with X as center. Intersection points
of the two arcs are points on the ellipse. Repeat this step for all
the remaining points. Use the French curve to connect the points,
thus drawing the ellipse.
Figure 7. Drawing an ellipse by arcs of circle method.
Constructing an Ellipse (Concentric Circle Method)
Concentric circle method is is used when the major axis and minor axis of the ellipse iis given. This method is illustrated in figure 8 and discussed below:
Figure 8. Concentric circle method of drawing ellipse
Drawing Tangent and Normal to any conic
When a tangent at any point on
the curve (P) is produced to meet the directrix, the line joining
the focus with this meeting point (FT) will be at right angle to
the line joining the focus with the point of contact
(PF).
The normal to the curve at any point
is perpendicular to the tangent at that point.
Figure 9. The method of drawing tangent and normal to any conic section at a particular point.
Parabola
Parabola
A parabola is obtained when a section plane, parallel to one of the generators cuts the cone. This is illustrated in figure 1.
Figure 1. Obtaining a parabola from a cone.
Parabola
(Applications)
There are a large number oif
applications for parabolic shapes. Some of these are in
searchlight mirrors, telescopic mirrors, a beam of uniform strength
in design applications, the trajectory of the weigtless flight,
etc. These are shown in figure 2.
Figure 2. Few applications of parabolic shapes.
Constructing a Parabola (Eccentricity Method)
The method of constructing a parabola by the eccentricity method where the distance of the focus from the directrix is 60 mm is shown in figure 3 and explained below.
Figure 3. Construction of parabola by eccentricity method.
Constructing a Parabola (Parallelogram Method)
Parabola can also be constructed
by parallelogram method. This is illustrated by the example below
and shown in figure 4.
Example: Draw a parabola of base 100
mm and axis 50 mm if the axis makes 70° to the base.
Figure 4. Construction of parabola by parallelogram method.
Tangent Method
This method can be used when the base and the axis, or base and the inclinations of tangents at open ends of the parabola with the base are given. The method is shown in figure 5.
This method can be used
when the base and the axis, or base and the inclinations of
tangents at open ends of the parabola with the base are given. The
method is shown in figure 5.
Draw the line AB representing the
base of the parabola.
Draw the Axis EF representing the
height of the parabola.
Produce EF to O such that EF =
OF
Join OA and OB
Divide OA and OB in to the
same number of parts say 8
Mark the division points as
shown
Draw lines joining 1 to 1’, 2 to 2’,
3 to 3’, etc.
Draw a curve starting from A
and tangent to the lines 11’, 22’, 33’, etc, which is the
required parabola.
Figure 5. Tangent method of drawing a parabola.
Hyperbola
Hyperbola
A Hyperbola is obtained when
a section plane, parallel/inclined to the axis cuts the cone on one
side of the axis. This is illustrated in figure
1.
A Rectangular Hyperbola is
obtained when a section, parallel to the axis cuts the cone on one
side of the axis.
Figure 1. illustration of a hyperbola.
Hyperbola (Applications)
Hyperbolic shapes finds large number of industrial applications like the shape of cooling towers, mirrors used for long distance telescopes, etc. (as shown in figure 2)
Figure 2. use of hyperbolic shapes in engineering applications. (source: internet)
Constructing a Hyperbola
(Eccentricity Method)
Construction of hyperbola by
eccentricity method is similar to ellipse and
parabola
Construction of a hyperbola
by eccentricity method is illustrated in figure 3, where the
eccentricity, e = 3/2 and the distance of the focus from the
directrix = 50 mm.
A hyperbola is mathematically defined as the set of points in a plane whose distances from two fixed points called foci, in the plane have a constant difference.
Constructing a
Hyperbola
Hyperbola can also be
construct if the distance between Foci and Distance between
vertices are known. This is illustrated in figure
4.
Draw the axis of
symmetry and construct a perpendicular through the axis.
Locate focal point F equidistant from the perpendicular and on
either side of it. Locate points A and B on the axis
equidistant from the
perpendicular.
AB is the distance between
vertices.
With F as center and radius
R1, and draw the arcs. With R1 + AB, radius, and F as center,
draw a second set of arcs. The intersection of the two arcs on each
side of the perpendicular are points on the
hyperbola.
Select a new radius R2 and repeat step 2. Continue this process until several points on the hyperbola are marked
Figure 4. Construction of a hyperbola
Spirals
Spirals
A spiral is a curve traced by a
point moving along a line in one direction, while the line is
rotating
in a plane about one of its ends or
any point on it.
e.g. Turbine casing, spiral casings.
etc. In other words it is the locus of a point which moves around a
centre, called the pole, while moving towards or away from
the centre.
The point which generates the
curve is called the generating point or tracing point.The point
will move along a line called the radius vector while the line
itself rotates about one of its end points.
Generally for engineering
applications two types of spirals are encountered. They
are:
Figure 1. A typical archemedian Spiral
Figure 2 A typical Logarithemic spiral
Archemedian
Spiral
The steps used to obtain an
Archemedian spiral is shown in figure 3 with the help of the
problem given below.
Problem: A point moves away
from the pole O and reaches a distance of 50 mm while moving around
it once.
Its movement from O is uniform
with its movement around. Draw the
curve.
Solution: Draw a circle with
diameter 50 mm and divide it into a number of equal segments, (say
six). Label the intersections between
the radius and the circle as points
1 through 6. Divide radius 06 into the same number of equal
parts (i.e. six).
Mark points on the radius as
1΄, 2΄, etc.
With O as the center, draw an
arc of radius 01΄, between 06 & 01. Mark the point of
intersection of the arc with radius 01.
Then draw an arc of radius
02΄, between 06 & 02. Repeat this process until arcs have
been drawn from all the points on the radius
06.
Using French curve, connect the
intersection points in the order, they were marked i.e. point on
01, point on 02 radius, point on 03….
Figure 3. Steps in drawing an Archemidian spiral
Logarithmic
spiral
In logarithmic spiral, the ratio of
the lengths of consecutive radius vectors enclosing equal angles
always remains constant. i.e. the values of vectorial angles are in
arithmetical progression . The corresponding values of
radius vectors arte in geometric progression.
The construction of a logarithmic
spiral is illustrated in figure 4 as solution to the
following problem.
Problem: Ratio of lengths of radius
vectors enclosing angle of 30° = 6:5. Final radius vector of the
spiral is 90 mm. Draw the spiral.
Solution:
Draw line AB and AC inclined at
30°.
On line AB, mark A12 = 90
mm.
A as center and A12 radius draw an
arc to cut AC at 12΄.
Mark A11 (= 5/6 of A12) on AB. Join
12΄ and 11.
Draw an arc with A as center and A11
radius to cut the line AC at 11΄.
Draw a line through 11΄
parallel to 12΄11 to cut AB at 10.
Repeat the procedure to obtain
points 9΄, 8΄, 7΄…0.
OP12 = A12΄, OP11 =
A11΄….
Figure 4. Logarithmic spiral
Normal and tangent to an
Archemedian spiral.
The normal to an Archemedian Spiral
at any point is the hypotenuse of the right angles triangle having
the
other two sides equal to the length
of the radius vector at that point and the constant of the
curve
The constant of the curve is equal
to the difference between the length of any two radii divided
by the
circular measure of the angle
between them.
The steps followed to draw the
normal and tangent to a spiral at any point N is illustrated in
figure 5.
Draw the radius vector
ON
Draw OM perpendicular to ON and
length equal to the constant of the curve.
Join MN
MN is the normal at point
N
Draw PQ perpendicular to MN to
obtain the tangent at N.
Figure 5. Drawing a tangent and normal to the spiral at any point.
Roulettes
Roulettes
Roulettes are curves generated by the rolling contact of one curve or line on another curve or line. There are various types of roulettes. The most common types of roulettes used in engineering practice are: Cycloids, trochoids, and Involutes. Assume a wheel is rolling along a surface without slipping. Trace the locus of a point on the wheel. Depending on the position of the point and the geometry of the surface on which the wheel rolls , different curves are obtained. Table 1 provides the general classification of roulettes.
Cycloid: Cycloid is
generated by a point on the circumference of a circle rolling along
a straight line without slipping.
Epicycloid: The cycloid is
called Epicycloid when the generating circle rolls along the
circumference of another circle outside it .
Hypocycloid: Hypocycloid is
obtained when the generating circle rolls along the circumference
of another circle but inside it.
Table 1 Classification of Cycloidal curve
Generating Circle 

On the directing line 
Outside the directing line 
Inside the directing line 

Generating point 
On the generating circle 
Cycloid 
Epicycloid 
Hypocycloid 
Outside the generating circle 
Superior trochoid 
Superior epitrochoid 
Superior Hypotrochoid 

Inside the generating circle 
Inferior trochoid 
Inferior epitrochoid 
Inferior hypotrochoid 
Cycloid
A Cycloid is generated by a point on
the circumference of a circle rolling along a straight line without
slipping.
The rolling circle is called
the Generating
circle
The straight line is called the
Directing line or Base line
Figure 1 illustrates the procedure for drawing a cycloid.
Generating circle has its center
at C and has a radius of CP’. Straight line PP’ is equal in length
to the circumference of the circle and is tangent to the circle at
point P’. Divide the circle into a number of equal segments, such
as 12. Number the intersections of the radii and the circle.
From each point of intersection on the circle, draw a
construction line parallel to line PP’ and extending up to
line P’C’. Divide the line CC’ into the same number of equal
parts, and number them. Draw vertical lines from each point to
intersect the extended horizontal centerline of the circle.
Label each point as C1, C2, C3, …. C12.
Using point C1 as the center and
radius of the circle CP’, draw an arc that intersects the
horizontal line extended from point 1 at P1. Set the compass
at point C2, then draw an arc that intersects the horizontal line
passing through point 2 at P2. Repeat this process using
points C3, C4, …. C12, to locate points along the horizontal
line extended from points 3, 4, 5, etc.. Draw a smooth curve
connecting P1, P2, P3, etc to form the cycloid
Epicycloid
Epicycloid is the curve generated by
a point on the circumference of a circle which rolls without
slipping along another circle outside it. This is illustrated in
figure 2.
Figure 2. Illustrates the generation of an epicycloid.
With O as centre and radius OP
(base circle radius), draw an arc PQ. The included angle θ = (r/R)
x 360°. With O as centre and OC as radius, draw an arc to represent
locus of centre.
Divide arc PQ in to 12 equal parts
and name them as 1’, 2’, …., 12’. Join O1’, O2’, … and produce them
to cut the locus of centres at C1, C2, ….C12. Taking C1 as centre,
and radius equal to r, draw an arc cutting the arc
through 1 at P1. Taking C2 as centre and with the same radius, draw
an arc cutting the arc through 2 at P2Similarly obtain points
P3, P3, …., P12. Draw a smooth curve passing through P1, P2….. ,
P12, which is the required epiclycloid.
Hypocycloid
Hypocycloid is the curve
generated by a point on the circumference of a circle which rolls
without slipping inside another circle.
The construction of a hypocycloid is illustrated in figure 3.
Figure 3 Construction of a hypocycloid.
With O as centre and radius OP
(base circle radius), draw an arc PQ. The included angle θ = (r/R)
x 360°. With O as centre and OC as radius, draw an arc to represent
locus of centre.
Divide arc PQ in to 12 equal parts
and name them as 1’, 2’, …., 12’. Join O1’, O2’, …, O12’ so
as to cut the locus of centres at C1, C2, ….C12. Taking C1 as
centre, and radius equal to r, draw an arc cutting
the arc through 1 at P1. Taking C2 as centre and with the same
radius, draw an arc cutting the arc through 2 at P2. Similarly
obtain points P3, P3, …., P12. Draw a smooth curve passing
through P1, P2….. , P12, which is the required
hypocycloid.
Trochoid
Trochoid is a curve generated by a point fixed to a circle as the circle rolls along a straight line. If the point is outside the rolling circle, the curve obtained is called an inferior trochoid and when outside the circle is called superior trochoid. Figure 4 illustrates an inferior trochoid and a superior trochoid.
Figure 4 illustrating the superior and inferior trochoids.
Construction of an inferior
trochoid.
The construction procedure for
obtaining an inferior trochoid is shown in figure 5. With centre C’
and radius R, draw a circle. From A, draw a horizontal
line AB = 2πR. Draw C’ C” parallel and equal to AB and
divide it in to 12 equal parts C1, C2, C3 ,…. C12. Draw the
generating point Q’ along C’A. With centre C’ and radius
C’Q’ draw a circle and divide the circumference in to 12 equal
parts and label tham as 1,2,3, ….,12. With C1, C2,
C3,etc., as the centres and radius equal to C’Q’, cut arcs on the
horizontal lines through 1,2,3,etc., to locate the points Q1,
Q2, Q3, etc. Join Q1, Q2, Q3, etc., to obtain the inferior
trochoid.
Figure 5. Construction of an inferior trochoid.
Construction of a Superior
Trochoid
The construction procedure for
obtaining a superior trochoid is shown in Figure 6. With centre C’
and radius R, draw a circle. From A, draw a horizontal
line AB = 2πR. Draw C’ C” parallel and equal to AB and
divide it in to 12 equal parts C1, C2, C3 ,…. C12. Draw the
generating point P’ along C’A. With centre C’ and radius
C’P’ draw a circle and divide the circumference in to 12 equal
parts and label them as 1,2,3, ….,12. With C1, C2,
C3,etc., as the centres and radius equal to C’P’, cut arcs on the
horizontal lines through 1,2,3,etc., to locate the points P1,
P2, P3, etc. Join P1, P2, P3, etc., to obtain the superior
trochoid.
Figure 5 Construction of a superior Trochoid.
Trochoids
Trochoids
Trochoid is a curve generated by a
point fixed to a circle, within or outside its circumference,
as the circle rolls along a straight line. If the point
is outside the rolling circle, the curve obtained is called
an inferior trochoid and when outside the circle is called superior
trochoid. Figure 1 illustrates an inferior trochoid and
a superior trochoid.
Epitrochoid
Epitrochoid is a curve generated by
a point fixed to a ircle (within or outside its circumference,
which rolls on the outside of another circle. If the point is
outside the rolling circle, the curve obtained is called an
inferior epitrochoid and when outside the circle, it is called
superior epitrochoid.
Hypotrochoid
Hypotroichoid is a curve generated
by a point fixed to a ircle (within or outside its circumference,
which rolls inside another circle. If the point is outside the
rolling circle, the curve obtained is called an inferior
hypotrochoid and when outside the circle, it is called superior
hypotrochoid.
Figure 1 illustrating the superior and inferior trochoids.
Construction of an inferior
trochoid.
The construction procedure for
obtaining an inferior trochoid is shown in figure 2. With centre C’
and radius R, draw a circle. From A, draw a horizontal
line AB = 2πR. Draw C’ C” parallel and equal to AB and
divide it in to 12 equal parts C1, C2, C3 ,…. C12. Draw the
generating point Q’ along C’A. With centre C’ and radius
C’Q’ draw a circle and divide the circumference in to 12 equal
parts and label tham as 1,2,3, ….,12. With C1, C2,
C3,etc., as the centres and radius equal to C’Q’, cut arcs on the
horizontal lines through 1,2,3,etc., to locate the points Q1,
Q2, Q3, etc. Join Q1, Q2, Q3, etc., to obtain the inferior
trochoid.
Figure 2. Construction of an inferior trochoid.
Construction of a Superior
Trochoid
The construction procedure for
obtaining a superior trochoid is shown in Figure 3. With centre C’
and radius R, draw a circle. From A, draw a horizontal
line AB = 2πR. Draw C’ C” parallel and equal to AB and
divide it in to 12 equal parts C1, C2, C3 ,…. C12. Draw the
generating point P’ along C’A. With centre C’ and radius
C’P’ draw a circle and divide the circumference in to 12 equal
parts and label them as 1,2,3, ….,12. With C1, C2,
C3,etc., as the centres and radius equal to C’P’, cut arcs on the
horizontal lines through 1,2,3,etc., to locate the points P1,
P2, P3, etc. Join P1, P2, P3, etc., to obtain the superior
trochoid.
Figure 3 Construction of a superior Trochoid.
Involutes
Involute
An Involute is a
curve traced by the free end of a thread unwound from a circle or a
polygon in such a way that the thread is always tight and
tangential to the circle or side of the polygon.Figure 1 shows the
involute of a circle.
Construction of Involute of circle
Figure 1. Construction of involute of a circle.
Involute of Regular Polygon (pentagon)
Figure 2 shows the construction of Involute of a regular pentagon. Draw the pentagon ABCDE. Extend line AE to P6 such that length EP6 is equal to 5 times AE. Extend line BA, CB, DC, and ED. With A as centre and radius equal to AE draw an arc to intersect the line BA extended at P1. Next with B as centre and radius equal to A1, draw an arc to intersect the line BA extended at P2. With C as centre and radius equal to A2, draw an arc to intersect the line DC extended at P3. The procedure is repeated till point P5 is obtained. Draw a smooth curve passing through P1, P2, P3, …, P5 to obtain the involute of the pentagon.
Figure 2. Involute of a pentagon.