Comprehensive Study Guide on Computer-Aided Engineering Graphics and Engineering Drawing Principles

UNIT I: INTRODUCTION TO AUTOCAD, SCALES, AND ENGINEERING CURVES

• Computer-Aided Design (CAD): CAD is the use of interactive graphics programs to develop 2D and 3D models, assemblies, part lists, and dimensional drawings of various components, structures, or objects with the help of a computer. Data generated by a CAD system can be directly utilized by a Computer-Aided Manufacturing (CAM) system.

• Advantages of CAD:

  • Reduction of drafting labor.

  • Direct cost savings.

  • High accuracy up to one millionth of a unit ($10^{-6}$ units).

  • Improvement in the general flow of information throughout an organization.

  • Rapid evaluation of alternative designs.

  • Efficient re-use of common parts across multiple products.

• Starting up AutoCAD:

  • Select AutoCAD from the Start menu (Start/Programs/AutoCAD) or double-click the AutoCAD desktop icon.

• AutoCAD Screen Layout:   

  

  • Title bar: Displays the application name and current drawing filename.

  • Menu bar: Provides access to drop-down menus for all commands.

  • Toolbars: Graphical icons representing drawing and editing tools.

  • Document window / Drawing area: The main graphical workspace.

  • Command window: Text interface for typing commands and receiving prompts.

  • Status bar: Displays coordinate readouts and status toggles.

• Drawing Limits (LIMITS):

  • Defines the extent of the drawing workspace and grid display area.

  • Can toggle Limits mode ON or OFF. When ON, points picked outside specified boundaries are rejected.

  • Default scale is full scale ($1:1$).

  • Example Command Sequence:

  • Command: limits

  • Lower left corner <0.0000, 0.0000>: 0,0

  • Upper right corner <12.0000, 9.0000>: 420,300 (Sets boundary to A3 paper size in mm).

• Grid (GRID) and Snap (SNAP):

  • Snap: Forces crosshairs to move in fixed increments along a regular grid. Toggled via $F9$ key or status bar. Default $X$ and $Y$ spacing is $1$ unit. Styles include Standard (for orthographic drawings) and Isometric (for isometric projections).

  • Grid: Displays a regular matrix of visual dots on screen acting as drawing paper background. Toggled via $F7$ key. For A3 size drawings, a grid spacing of $10$ units is typical.

  • Example Command Sequence:

  • Command: grid -> Spacing: 10

  • Command: snap -> Spacing: 10

  • Command: zoom -> All (Scales display to fit defined drawing limits).

• Units (UNITS):

  • Controls precision, measurement unit style, and angle measurement formats.

  • Unit Measurement Formats:

  1. Scientific (e.g., $1.55 \times 10^1$)

  2. Decimal (e.g., $15.50$)

  3. Engineering (e.g., $1'-3.50"$)

  4. Architectural (e.g., $1-3/4"$)

  5. Fractional (e.g., $15 1/2$)

  • Angle Measurement Formats:

  1. Decimal degrees (e.g., $45.0000^\circ$)

  2. Degree/minute/second (e.g., $45^\circ 0' 00"$)

  3. Grads (e.g., $50.0000g$)

  4. Radians (e.g., $0.7854r$)

  5. Surveyor's unit (e.g., $N 45^\circ 0' 0" E$)

• Keyboard Function Keys:

  • $F1$ / $F2$: Graphical screen / Text screen toggle and Help menu.

  • $F6$: Coordinate display mode toggle.

  • $F7$: Grid mode toggle.

  • $F8$: Ortho (orthogonal) mode toggle (restricts cursor to vertical/horizontal movements).

  • $F9$: Snap mode toggle.

• Coordinate Systems:

  • Absolute Coordinates: Locations specified by precise distance from origin $(0,0)$.

  • Example: Line from $0,0$ to $-3,4$ to $4,2$.

  • Relative Coordinates: Points specified relative to the previous point using the @ symbol.

  • Example: Line from $0,0$ to @-3,4 to @7,-2.

  • Polar Coordinates: Points defined by distance and angle relative to current location (@distance<angle). Angle $0^\circ$ points due east (right).   

    

• AutoCAD Draw Commands:

  • LINE: Draws straight segments between points. Options: U (Undo last segment), C (Close polygon to initial starting point).

  • CIRCLE: Defined via Center & Radius, Center & Diameter, 2-Point, 3-Point, or Tangent-Tangent-Radius (TTR).

  • ARC: Defined via 3-point, Start-Center-End, Start-Center-Angle, Start-Center-Length, Start-End-Angle, Start-End-Radius, or Start-End-Direction.

  • POINT: Creates a point object; style and size adjusted via Settings -> Point Style.

  • POLYLINE (PL): Single object composed of joined straight and/or arc segments.

  • DONUT: Solid filled ring defined by inner diameter and outer diameter.   

    

  • RECTANGLE: Draws a rectangle defined by two diagonal corner coordinates.

  • POLYGON: Creates regular 2D polygons ($3$ to $1024$ sides). Options: Edge (E), Inscribed inside circle (I), Circumscribed around circle (C).

  • ELLIPSE: Created via Axis-Eccentricity, Center-Axis-Axis, or Rotation angle.

  • HATCH: Fills enclosed areas with crosshatch patterns representing specific materials or surfaces.

• Object Snap (OSNAP):   

  

  • Precision tool that locks onto geometric points (Endpoint, Midpoint, Center, Geometric Center, Node, Quadrant, Intersection, Extension, Insertion, Perpendicular, Tangent, Nearest, Apparent Intersection, Parallel).

  • Activated running-mode or temporary overrides (Shift + Right Click or typing 3-letter codes e.g., CEN).

• AutoCAD Modify & Edit Commands:   

  

  • ERASE (E): Removes objects.

  • UNDO (U): Reverses last command.

  • MOVE (M): Relocates objects from base point to destination.

  • COPY (CO/CP): Duplicates objects; use M option for multiple copies.

  • MIRROR (MI): Creates symmetrical reflection across a defined line.

  • ROTATE (RO): Rotates objects around a base point (default counterclockwise).

  • OFFSET (O): Creates parallel lines or concentric curves at specified distance.

  • EXTEND (EX): Lengthens objects to meet a boundary edge.

  • TRIM (TR): Cuts objects at intersections with cutting edges.

  • BREAK (BR): Removes a section of an object between two points.

  • FILLET (F): Creates rounded arc corners between two non-parallel lines.

  • CHAMFER (CHA): Creates angled bevels between intersecting lines.

  • STRETCH: Elongates or contracts objects using crossing selection windows.

  • ARRAY (AR): Copies objects in Rectangular or Polar (circular) patterns.

  • SCALE (SC): Proportionally alters object sizes based on scale factor.

• Object Selection Options:

  • Pickbox (single object), Window (W - entirely inside), Crossing (C - inside or touching), Previous (P), Remove (R), Add (A), Last (L), Fence (F - line path), WindowPolygon (WP), CrossingPolygon (CP), All.

• Display Commands:

  • ZOOM (Z): All (A), Center (C), Dynamic (D), Extend (E), Left (L), Previous (P), Window (W), Scale (X/XP).

  • PAN: Shifts viewing window without altering object coordinates or magnification.

  • REDRAW (R): Refreshes screen display and cleans temporary blip marks.

  • VIEWRES: Adjusts curve generation smoothness/resolution for arcs and circles.

• Text and Dimensioning:

  • TEXT (DT): Adds single-line text; styled via STYLE command.

  • DIMENSION: Linear (Horizontal, Vertical, Aligned, Continuous, Baseline, Rotated), Radial, Angular, Leader, Ordinate. Managed via Dimension Style Manager dialog.   

    

• Engineering Graphics Fundamentals:

  • Drawing Instruments: Board, Mini-Drafter (combines functions of T-square, set-squares, protractor, scales), Instrument box (Compass, Dividers, Inking Pens).

  • Compass Lead Adjustment: Pencil lead must extend $1\,mm$ above compass point tip to compensate for tip insertion into the board.   

    

  • Pencil Lead Grades:

  • HB: Soft grade for borderlines, lettering, and freehand sketching.

  • H: Medium grade for visible outlines, edges, and boundary lines.

  • 2H: Hard grade for construction lines, dimension lines, center lines, hatching lines, extension lines, and hidden lines.

  • Drawing Sheet Sizes:

  • Standard ratio: $x : y = 1 : \sqrt{2}$ and $x \times y = 1\,m^2$ for $A0$.     

    

  • $A0$: $841\,mm \times 1189\,mm$

  • $A1$: $594\,mm \times 841\,mm$

  • $A2$: $420\,mm \times 594\,mm$ (Preferred for classroom drafting)

  • $A3$: $297\,mm \times 420\,mm$

  • $A4$: $210\,mm \times 297\,mm$

  • Title Block:

  • Positioned at bottom right corner of sheet.

  • Maximum dimensions: Length $170\,mm$, Width $65\,mm$.

  • Contains: Title, Drawing number, Scale, Projection symbol, Name of firm, Staff initials.

  • Line Conventions:     

    

  • Continuous wide line ($0.7\,mm$): Visible outlines, edges, borders.

  • Continuous narrow line: Dimension, extension, leader, construction, hatching lines.

  • Continuous narrow freehand / zigzag lines: Short/long break lines.

  • Dashed narrow line: Hidden edges and outlines ($3\,mm$ dashes, $1\,mm$ gaps).

  • Long-dashed dotted narrow line: Center lines, axes of symmetry, cutting planes (thickened at ends).

  • Material Conventions:     

    

• Conic Sections:

  • Formed by intersection of a cone (apex angle $2\theta$) with a cutting plane at angle $\alpha$ to the axis:

  • Circle: Cutting plane perpendicular to axis ($\alpha = 90^\circ$).

  • Ellipse: Cutting plane inclined to axis at angle greater than semi-apex angle ($\theta < \alpha < 90^\circ$).

  • Parabola: Cutting plane parallel to slant generator ($\alpha = \theta$).

  • Hyperbola: Cutting plane at angle less than semi-apex angle ($0^\circ \le \alpha < \theta$).

  • Eccentricity ($e$): Ratio of distance from focus to distance from directrix ($e = \frac{VF}{VA}$).

  • Parabola: $e = 1$

  • Ellipse: $e < 1$

  • Hyperbola: $e > 1$

• Construction of Conics (Eccentricity Method):

  • Parabola ($e=1$) (Focus $50\,mm$ from directrix):     

    

  1. Axis $AB$ perpendicular to directrix $CD$.

  2. Mark Focus $F$ at $50\,mm$ from directrix $A$.

  3. Locate vertex $V$ at midpoint of $AF$ ($AV = VF = 25\,mm$) so $e = \frac{VF}{VA} = 1$.

  4. Draw $VE \perp AB$ with $VE = VF$. Join $AE$ and extend.

  5. Mark points $1, 2, 3$ along axis; draw perpendiculars meeting $AE$ extended at $1', 2', 3'$.

  6. Arcs centered at $F$ with radii $1-1', 2-2'$ intersect vertical lines to form parabola curve.

  7. Tangent & Normal at point $M$ ($40\,mm$ from directrix): Join $MF$. Draw $FT \perp MF$ meeting directrix at $T$. Line $TM$ extended is tangent; perpendicular $NN$ at $M$ is normal.

  • Ellipse ($e=2/3$) (Focus $50\,mm$ from directrix):     

    

  1. Divide $AF = 50\,mm$ into $5$ equal divisions ($10\,mm$ each).

  2. Mark vertex $V$ at 3rd division from $A$ ($VA = 30\,mm, VF = 20\,mm$) yielding $e = \frac{2}{3}$.

  3. Draw $VE = VF = 20\,mm$ perpendicular to axis. Join $AE$ and extend.

  4. Mark points on axis, erect perpendiculars to $AE$ line, draw arcs centered at $F$ with radius equal to vertical ordinate heights to plot closed ellipse curve.

  • Hyperbola ($e=3/2$) (Focus $50\,mm$ or $65\,mm$ from directrix):     

    

  1. For $AF = 65\,mm$, divide into $5$ parts. Vertex $V$ located at 2nd division from $A$ ($VA = 26\,mm, VF = 39\,mm$) yielding $e = \frac{3}{2}$.

  2. Draw $VE = VF$, connect $AE$ extended, plot points using arcs centered at $F$ with radii equal to ordinates.

• Engineering Scales:

  • Representative Fraction ($RF$): Ratio of drawing length to actual length:     $RF = \frac{\text{Length of Drawing}}{\text{Actual Length of Object}}$

  • Length of Scale ($L_s$):     $L_s = RF \times \text{Maximum Length to be measured}$

  • Types of Graphical Scales: Plain Scales, Diagonal Scales, Vernier Scales, Comparative Scales, Scale of Chords.

• Plain Scale Solved Calculations:

  • Example 1: Scale $1:40$ to read meters and decimeters up to $6\,m$. Distance 4.7\,m$.\n - RF = \frac{1}{40}\n - L_s = \frac{1}{40} \times 6\,m = \frac{1}{40} \times 6000\,mm = 150\,mm\n - Divide 150\,mm$rectangle into$6$main divisions ($1\,m$each). Divide 1st division into$10$subdivisions ($1\,dm each).\n - Mark 4.7\,m$($4\,m$right of zero,$7\,dm left of zero).\n    \n - *Example 2*: Scale 1:5$for decimeters and centimeters up to$1\,m$. Show$7.6\,dm\n - RF = \frac{1}{5}\n - L_s = \frac{1}{5} \times 1000\,mm = 200\,mm\n - Divide into 10$parts ($1\,dm$each). First part divided into$10$subunits ($1\,cm$each). Mark$7\,dm$right,$6\,cm left.\n - *Example 3*: 1\,cm$map line =$4\,m$actual. Max length$50\,m$. Mark$44\,m$.

  • $RF = \frac{1\,cm}{400\,cm} = \frac{1}{400}$

  • $L_s = \frac{1}{400} \times 50000\,mm = 125\,mm$

  • Divide into $5$ parts ($10\,m$ each). First part into $10$ subunits ($1\,m$ each).

  • Example 4: Area $49\,cm^2$ represents $16\,m^2$. Max length $8\,m$. Mark $6\,m\,9\,dm$

  • $RF = \sqrt{\frac{49\,cm^2}{16\,m^2}} = \frac{7\,cm}{4\,m} = \frac{7}{400}$

  • $L_s = \frac{7}{400} \times 8000\,mm = 140\,mm$

  • Example 5: $1.5\,in = 1\,ft$. Max length $4\,ft$. Mark $2\,ft\,10\,in$

  • $RF = \frac{1.5}{12} = \frac{1}{8}$

  • $L_s = \frac{1}{8} \times 4 \times 12 \times 25.4\,mm = 152.4\,mm$

• Diagonal Scale Solved Calculations:

  • Used to measure three consecutive units (e.g., meters, decimeters, centimeters).

  • Example 1: $RF = 1/50$, max length $5\,m$. Mark 4.68\,m$.\n - L_s = \frac{1}{50} \times 5000\,mm = 100\,mm\n - Rectangle 100\,mm \times 50\,mm$. Divide length into$5$parts ($1\,m$each). Divide 1st division into$10$parts ($1\,dm$each) with diagonal lines. Divide vertical height into$10$parts ($1\,cm each).\n - Mark 4.68\,m$:$4\,m$right of zero,$6\,dm left along diagonal, elevated to 8th horizontal line.\n    \n - *Example 2*: RF = 1/6250$, max length$1\,km$. Mark$653\,m$.

  • $L_s = \frac{1}{6250} \times 10^6\,mm = 160\,mm$

  • Example 3: $RF = 1/32$, max length $4\,yards$. Mark 1\,yd\,2\,ft\,7\,in$.\n - L_s = \frac{1}{32} \times 4 \times 3 \times 12 \times 25.4\,mm = 114.3\,mm\n - *Example 4*: Room volume 1728\,m^3$represented by$4\,cm$cube side. Max length$50\,m$. Mark$37.6\,m$.

  • $RF = \frac{4\,cm}{\sqrt[3]{1728}\,m} = \frac{4\,cm}{12\,m} = \frac{4\,cm}{1200\,cm} = \frac{1}{300}$

  • $L_s = \frac{1}{300} \times 50000\,mm = 166.67\,mm$

• Cycloidal Curves:

  • Cycloid: Curve traced by a point on the circumference of a circle rolling without slipping along a straight line.     

    

  • Construction: Rolling circle radius $R$. Base line length = circumference $2\pi R$. Divide circle and baseline into $12$ equal parts. Center line $OB$ contains center positions $O_1, O_2, \dots, O_{12}$. Draw horizontal lines through circle division points; draw arcs centered at $O_n$ with radius $R$ intersecting corresponding horizontal lines.

  • Epicycloid: Curve traced by a point on a circle rolling on the outside of a directing circle (radius $R$).

  • Subtended angle formula: $\theta = \frac{r}{R} \times 360^\text{o}$

  • Hypocycloid: Curve traced by a point on a circle rolling on the inside of a directing circle (radius $R$).

  • Example: Diameter $40\,mm$ ($r=20\,mm$) rolling inside $200\,mm$ diameter circle ($R=100\,mm$).

  • $\theta = \frac{20}{100} \times 360^\text{o} = 72^\text{o}$.

UNIT II: PROJECTIONS OF POINTS AND STRAIGHT LINES

• Projections of Points:   

  

  • Standard Notation Rules:

  • Actual spatial point: Capital letters ($A, B, C$).

  • Front View (FV) projected on Vertical Plane (VP): Lowercase with dash ($a', b', c'$).

  • Top View (TV) projected on Horizontal Plane (HP): Lowercase without dash ($a, b, c$).

  • Side View (SV) projected on Profile Plane (PP): Lowercase with double dash ($a'', b'', c''$).

  • Reference line: $XY$. Projectors connecting FV and TV are always perpendicular to $XY$.

  • Quadrant Positions & Conversions:

  • Point $P$ ($30\,mm$ above HP, $40\,mm$ in front of VP): 1st Quadrant $a'$ is $30\,mm$ above $XY$, $a$ is $40\,mm$ below $XY$.

  • Point $Q$ ($25\,mm$ above HP, $35\,mm$ behind VP): 2nd Quadrant both $q'$ ($25\,mm$) and $q$ ($35\,mm$) lie above $XY$.

  • Point $R$ ($32\,mm$ below HP, $45\,mm$ behind VP): 3rd Quadrant $r'$ is $32\,mm$ below $XY$, $r$ is $45\,mm$ above $XY$.

  • Point $S$ ($35\,mm$ below HP, $42\,mm$ in front of VP): 4th Quadrant both $s'$ ($35\,mm$) and $s$ ($42\,mm$) lie below $XY$.

  • Point $T$ (in HP, $30\,mm$ behind VP): $t'$ on $XY$, $t$ is $30\,mm$ above $XY$.

  • Point $U$ (in VP, $40\,mm$ below HP): $u'$ is $40\,mm$ below $XY$, $u$ on $XY$.

  • Point $V$ (in VP, $35\,mm$ above HP): $v'$ is $35\,mm$ above $XY$, $v$ on $XY$.

  • Point $W$ (in HP, $48\,mm$ in front of VP): $w'$ on $XY$, $w$ is $48\,mm$ below $XY$.

• Projections of Straight Lines:

  • True Length (TL): Actual actual physical length of the line segment.

  • Orientations:

  • Parallel to both HP and VP: FV ($a'b'$) and TV ($ab$) both show True Length parallel to $XY$.

  • Perpendicular to HP, parallel to VP: FV ($a'b'$) shows True Length vertical; TV ($ab$) is a single point.

  • Perpendicular to VP, parallel to HP: FV ($a'b'$) is a point; TV ($ab$) shows True Length vertical/perpendicular to $XY$.

  • Inclined to HP at angle $\theta$, parallel to VP: FV shows True Length at angle $\theta$ to $XY$; TV is a shortened horizontal line parallel to $XY$.

  • Inclined to VP at angle $\phi$, parallel to HP: TV shows True Length at angle $\phi$ to $XY$; FV is a shortened line parallel to $XY$.

  • Inclined to both reference planes: Both FV and TV appear foreshortened at apparent angles $\alpha$ and $\beta$ respectively.

• Traces of Straight Lines:

  • Horizontal Trace (HT): Intersection point of line (or extended line) with Horizontal Plane. Found by extending FV to intersect $XY$ at $h$, then projecting vertically to intersect TV line extended at $HT$.

  • Vertical Trace (VT): Intersection point of line with Vertical Plane. Found by extending TV to intersect $XY$ at $v$, then projecting vertically to intersect FV line extended at $VT$.

• Comprehensive Solved Problems (Lines):

  • Problem 1: Line $AB$ ($75\,mm$ long), TV measures $65\,mm$, FV measures $50\,mm$. End $A$ in HP and $12\,mm$ in front of VP. Determine inclinations.

  • Solution procedure: Plot $a'$ on $XY$ and $a$ $12\,mm$ below $XY$. With radius $75\,mm$ and center $a$, draw true length arc to locus line of $b$ to determine $\phi$. Use given FV length ($50\,mm$) and TV length ($65\,mm$) to construct projections and measure true angles $\theta$ and $\phi$.

  • Problem 2: End projectors of line $AB$ are $50\,mm$ apart. $A$ is $20\,mm$ above HP, $30\,mm$ in front of VP. $B$ is $10\,mm$ below HP, $40\,mm$ behind VP. Determine True Length and Traces.

  • Solution procedure: Plot projector line $A$ ($a'$ $20\,mm$ above, $a$ $30\,mm$ below $XY$). Draw projector line $B$ $50\,mm$ away ($b'$ $10\,mm$ below, $b$ $40\,mm$ above $XY$). Connect $a'b'$ (FV) and $ab$ (TV). Rotate projections to horizontal loci to obtain True Length and measure true angles $\theta, \phi$ and location of $HT$ and $VT$.

UNIT III: PROJECTION OF PLANES

• Fundamentals of Planes:   

  

  • Two-dimensional surfaces with zero thickness (Square, Rectangle, Circle, Pentagon, Hexagon, Triangle).

  • Types of Planes:

  1. Perpendicular Planes: Surface perpendicular to one reference plane and parallel or inclined to the other.

  2. Oblique Planes: Surface inclined to both reference planes.

  • Traces of Planes:

  • Horizontal Trace (HT): Line of intersection of plane surface with HP.

  • Vertical Trace (VT): Line of intersection of plane surface with VP.

• Standard Orientations of Plane Surfaces:

  • Parallel to HP & perpendicular to VP: TV shows true shape; FV is a line parallel to $XY$.

  • Parallel to VP & perpendicular to HP: FV shows true shape; TV is a line parallel to $XY$.

  • Inclined to HP at angle $\theta$ & perpendicular to VP: FV is a line inclined at $\theta$ to $XY$; TV is a foreshortened shape.

  • Inclined to VP at angle $\phi$ & perpendicular to HP: TV is a line inclined at $\phi$ to $XY$; FV is a foreshortened shape.

  • Perpendicular to both HP and VP: Both FV and TV are lines perpendicular to $XY$; Side View shows true shape.

• Three-Step Method for Planes Inclined to Both Planes:

  • Step 1 (Initial Position): Assume plane surface parallel to HP (if surface is inclined to HP) or parallel to VP (if surface inclined to VP). Draw view showing True Shape first, ensuring edge/side making angle is perpendicular to $XY$ line.

  • Step 2 (Surface Inclination): Tilt the line-view in the second projection to the required surface inclination angle ($\theta$ or $\phi$) and project the modified foreshortened view.

  • Step 3 (Edge/Side Inclination): Rotate the foreshortened view or edge in the third projection to satisfy the edge inclination angle and project final true front and top views.

• Solved Plane Problems:

  • Square Plane ($40\,mm$ side): Parallel to HP, $20\,mm$ above HP. FV is line $20\,mm$ above $XY$; TV is square $40\,mm$.

  • Pentagonal Plane ($30\,mm$ side): Edge on HP, surface inclined $45^\text{o}$ to HP. Initial TV is true pentagon with vertical edge. Secondary FV is line inclined at $45^\text{o}$ to $XY$.

  • Hexagonal Plate ($30\,mm$ side): Surface perpendicular to HP, inclined $45^\text{o}$ to VP, one edge parallel to and $15\,mm$ in front of VP. Initial FV true hexagon; secondary TV line at $45^\text{o}$ to $XY$.

  • Circular Plane ($60\,mm$ diameter): Resting on VP point, center $40\,mm$ above HP, surface inclined $45^\text{o}$ to VP, perpendicular to HP. TV is line at $45^\text{o}$ to $XY$; FV is an ellipse.

  • Semi-Circle ($100\,mm$ diameter): Freely suspended from point on straight edge $30\,mm$ from midpoint, surface $45^\text{o}$ to VP. Centroid line suspended vertically in initial FV; tilted $45^\text{o}$ in secondary projections.

UNIT IV: PROJECTIONS OF SOLIDS AND DEVELOPMENT OF SURFACES

• Classification of Solids:

  • Polyhedra: Solids bounded by plane polygonal surfaces (faces).

  • Regular Polyhedra: Tetrahedron ($4$ equilateral triangles), Cube/Hexahedron ($6$ squares), Octahedron ($8$ equilateral triangles), Dodecahedron ($12$ regular pentagons), Icosahedron ($20$ equilateral triangles).

  • Prisms: Polyhedra with two equal parallel polygonal bases connected by rectangular faces. Named after base shape (Triangular, Square, Pentagonal, Hexagonal).     

    

  • Pyramids: Polyhedra with one polygonal base and triangular faces meeting at a single apex.

  • Solids of Revolution: Generated by revolving a 2D plane figure around an axis.

  • Cylinder (rectangle rotated around axis), Cone (right triangle rotated around altitude), Sphere (semicircle rotated around diameter).

• Projections of Solids - Solved Cases:

  • Axis perpendicular to HP: TV shows true base shape; FV shows height rectangle or triangle.

  • Example: Square pyramid ($40\,mm$ base, $60\,mm$ height) resting on base in HP.

  • Axis perpendicular to VP: FV shows true base shape; TV shows height profile.

  • Example: Pentagonal prism ($30\,mm$ base, $60\,mm$ length) base in VP.

  • Axis inclined to HP, parallel to VP:

  • Example: Hexagonal prism ($30\,mm$ base, $75\,mm$ axis) base edge on HP, axis $45^\text{o}$ to HP. Step 1: Draw prism vertical to HP. Step 2: Tilt FV line profile by $45^\text{o}$ to $XY$ and project secondary TV.

  • Axis inclined to VP, parallel to HP:

  • Example: Hexagonal prism ($30\,mm$ base, $65\,mm$ axis) base edge in VP, axis $30^\text{o}$ to VP. Step 1: Axis perpendicular to VP. Step 2: Tilt TV by $30^\text{o}$ to $XY$ and project secondary FV.

• Development of Surfaces:

  • Unfolding the 3D lateral boundary surface of an object flat onto a single 2D plane.

  • Methods of Development:

  1. Parallel Line Method: Used for prisms and cylinders (all generators/edges are parallel).

  2. Radial Line Method: Used for pyramids and cones (slant edges/generators converge at apex).

  3. Triangulation Method: Used for transition pieces and oblique solids.

  4. Approximation Method: Used for doubly curved surfaces like spheres.

• Development Solved Examples:

  • Truncated Pentagonal Prism (Base side $30\,mm$, height $70\,mm$, cut at $45^\text{o}$ bisecting axis):

  • Parallel line method. Stretch-out line length = Perimeter = $5 \times 30\,mm = 150\,mm$.

  • Divide stretch-out rectangle into $5$ equal segments of $30\,mm$. Project cut points from FV onto corresponding vertical edge lines and connect with straight segments.

  • Truncated Cylinder (Base diameter $50\,mm$, height $70\,mm$, cut at $45^\text{o}$ from top generator):

  • Parallel line method. Stretch-out line length = $\pi D = \pi \times 50\,mm = 157.08\,mm$.

  • Divide baseline into $12$ equal divisions. Project cut points from FV to form a smooth continuous curve using SPLINE tool.

  • Truncated Cone (Base diameter $50\,mm$, altitude $60\,mm$, cut at $45^\text{o}$ bisecting axis):

  • Radial line method. Base radius $r = 25\,mm$.

  • Slant height $R = \sqrt{r^2 + h^2} = \sqrt{25^2 + 60^2} = \sqrt{4225} = 65\,mm$.

  • Sector subtended angle:       $\theta = \frac{r}{R} \times 360^\text{o} = \frac{25}{65} \times 360^\text{o} = 138.46^\text{o} \approx 138^\text{o}$

  • Draw sector radius $65\,mm$, angle $138^\text{o}$, divide into $12$ radial parts, project true lengths of cut points onto outer true generator, draw curved cut profile.

  • Truncated Square Pyramid (Base side $40\,mm$, altitude $60\,mm$, cut at $45^\text{o}$ bisecting axis):

  • Radial line method. Determine true slant edge length $O'H'$ by rotating corner diagonal horizontal.

  • Radius $R = O'H'$. Layout $4$ isosceles triangles with base width $40\,mm$. Transfer true distances of cut points from apex to layout edges.

UNIT V: ISOMETRIC PROJECTIONS AND ORTHOGRAPHIC PROJECTIONS

• Isometric Projection Principles:   

  

  • Pictorial view showing three dimensions simultaneously.

  • Isometric Axes: Three lines meeting at a point making equal angles of $120^\text{o}$ to each other.

  • Isometric Lines: Lines parallel to isometric axes.

  • Non-Isometric Lines: Lines not parallel to isometric axes (e.g., diagonals, pyramid slant edges).

  • Isometric Planes: Planes parallel to the three principal planes of the isometric cube.

• Isometric Scale:   

  

  • Objects viewed in isometric projection undergo foreshortening due to spatial tilting.

  • Reduction ratio:     $\text{Isometric Length} = \sqrt{\frac{2}{3}} \times \text{True Length} \approx 0.815 \times \text{True Length} \approx \frac{9}{11} \times \text{True Length}$

  • Isometric View vs Isometric Projection:

  • Isometric View: Drawn using actual True Dimensions ($1:1$ scale).

  • Isometric Projection: Drawn using reduced Isometric Dimensions ($0.815$ scale).

• Isometric Construction Methods:

  • Circle in Isometric (Four-Center Method):

  1. Enclose circle of diameter $D$ in an isometric rhombus of side $D$.

  2. Mark midpoints $1, 2, 3, 4$ of the rhombus sides.

  3. Draw lines from obtuse corners to opposite side midpoints. The intersections define $4$ centers.

  4. Swing arcs from obtuse corners and intersection centers to form a smooth ellipse representation.

  • Non-Isometric Lines Methods:

  • Box Method: Solid enclosed in a rectangular bounding box; contact coordinates mapped along box edges.

  • Offset Method: Coordinates located by measuring perpendicular offset distances along isometric axes.

• Isometric Solved Solids & Composites:

  • Frustum of Hexagonal Pyramid: Base $35\,mm$, top $20\,mm$, height $80\,mm$. Constructed inside concentric isometric boxes at base and top heights.

  • Composite Solid (Sphere on Cube): Cube $70\,mm$ side, $30\,mm$ thick; Sphere $60\,mm$ diameter centered on top. Note: Isometric projection of a sphere is always a true circle with radius equal to True Radius ($R = 30\,mm$), centered at an isometric elevation of $0.815 \times R$ above the top face center.

• Orthographic Projection Fundamentals:

  • Projectors are parallel to each other and perpendicular to the projection plane.

  • Reference Planes: Vertical Plane (VP) and Horizontal Plane (HP) intersecting at Reference Line $XY$.

  • Quadrant Positions:     

    

  • First Angle Projection:

    • Object placed in 1st Quadrant (between observer and plane).

    • Front View lies ABOVE reference line $XY$.

    • Top View lies BELOW reference line $XY$.

    • Left Side View drawn on RIGHT side of Front View.

  • Third Angle Projection:

    • Object placed in 3rd Quadrant (plane placed between observer and object).

    • Front View lies BELOW reference line $XY$.

    • Top View lies ABOVE reference line $XY$.

    • Left Side View drawn on LEFT side of Front View.

• 3D Isometric to 2D Orthographic Conversion Procedure:

  1. Identify Front View direction (arrow $X$) to establish overall Width ($W$) and Height ($H$).

  2. Determine Top View looking from above to establish Width ($W$) and Depth ($D$).

  3. Project Side View from left/right to establish Depth ($D$) and Height ($H$).

  4. Draw visible edges as thick continuous lines ($0.7\,mm$) and hidden internal features as dashed narrow lines.