A cone base 75 mm diameter and axis

A cone base 75 mm diameter and axis 100 mm long, has its base on the HP. A section plane parallel to one of the end generators and perpendicular to the FP cuts the cone intersecting the axis at a point 75 mm from the base. Draw the sectional Top View and the true shape of the section • The section plane is parallel to one of the end generators and perpendicular to the frontal plane • It is therefore drawn in the Front View • It cuts the axis at a point 75 mm from the base as shown in Front View True shape of section (parabola) • Draw horizontal circles around the cone surface with center coinciding with the axis in the TV T F • Project corresponding points of intersection of the circles with the section plane in FV to the TV 100 75 75 Axis Section plane Join these points to get the section face For true shape of the section, draw an auxiliary view with reference line PRIMARY AUXILIARY parallel to the section plane VIEW

A pentagonal pyramid (side of base = 50 mm and height = 100 mm) is resting on its base on the ground with axis parallel to frontal plane and perpendicular to the top plane. One of the sides of the base is closer and parallel to the frontal plane. A vertical section plane cuts the pyramid at a distance of 15 mm from the axis with section plane making an angle of 50 o with FP. Draw the remaining part of the pyramid and the true shape of the cut section d e T p o a r c The plane is perpendicular to the top plane, therefore the section line is drawn in the Top View Section plane It cuts the base at f and j n m 50 o It cuts the edges at g and h b o’ Join these points to o form the section face n 1 p 1 F 15 1 p’ m 100 r 1 n’ m’ e’ a’ d’ b’ r’ c’ 50 The true shape of the section is drawn as an auxiliary view to the top view with the reference line parallel to the section plane

The pyramid is also cut by another plane that is perpendicular to the frontal plane, inclined at 70 o to the top plane and cuts the axis of the pyramid at 15 mm from the apex. Draw the projections of the remaining part of the pyramid and the true shape of the cut section d Since the section plane is perpendicular to the frontal plane, the section line is drawn in the front view The cutting plane cuts the axis of the pyramid (light blue) 15 mm below the apex e h j b T 70 o F o’ j 1 k’ j’ It also cuts the edges at h, j, l and k Project these points in the top view and join them k a k 1 l 1 True length c l g Parallel It cuts the base at g and i i 15 l’ h 1 h’ 100 Eliminate the edge oe and part of the edge oa which are cut off Project an auxiliary view of the True i 1 g 1 shape of the section by taking the reference line parallel to the section line g’, i’ e’ a’ d’ b’ c’ Axis of pyramid Section plane

d How to locate the point “l” Draw an imaginary horizontal line from the axis (light blue) to the edge oc intersecting at z l o y Project the point z into the Top view (oz is TL here) With o as center and oz as radius draw an arc cutting od at I This can also be done by projecting onto ob at y and rotating. Basically the imaginary line with length oz = oy is rotating inside the pyramid from one edge to another c z b o l y z This can also be obtained by drawing a line from z in the Top view parallel to dc (as dc is TL here) d b c

CONIC SECTIONS ELLIPSE, PARABOLA AND HYPERBOLA ARE CALLED CONIC SECTIONS BECAUSE THESE CURVES APPEAR ON THE SECTION OF A CONE WHEN IT IS CUT BY SOME TYPICAL CUTTING PLANES. Section Plane Through Generators Par abo la Ellipse Section Plane Parallel to end generator. Section Plane Parallel to Axis. Hyperbola 5

COMMON DEFINITION OF ELLIPSE, PARABOLA & HYPERBOLA: These are the loci of points moving in a plane such that the ratio of it’s distances from a fixed point And a fixed line always remains constant. The Ratio is called ECCENTRICITY. (E) A) For Ellipse E<1 B) For Parabola E=1 C) For Hyperbola E>1 6

Ellipse Found where? Arches, bridges, dams, monuments, man -holes, glands Equation: a: Half length of major axis b: Half length of minor axis Eccentricity: e < 1 Sum of distances of a point on the ellipse to the foci is constant C P PF 1 + PF 2 = constant = AF 1 + AF 2 = AB = Length of Major axis A CF 1 + CF 2 = 2 CF 1= AB/2 Minor axis B F 1 F 2 D Focus 7

Construct an ellipse, length of major and minor axis given: Arcs of circles method P Q RC 1 2 3 A B F 1 F 2 D PRINCIPLE: Sum of distances of point to foci = Length of major axis F 1 P = A 1, F 2 P = B 1 F 1 Q = A 2, F 2 Q = B 2 F 1 R = A 3, F 2 R = B 3 8

ELLIPSE Draw an ellipse, focus is 50 mm from the directrix and the eccentricity is 2/3 DIRECTRIX-FOCUS METHOD DIRECTRIX A C VE = VF 1 2’ 1’ P 2 E F 1 -P 1=F 1 -P 1’ = 1 -1’ P 1 F 1 -P 1/(P 1 to directrix AB) = F 1 ( focus) 1 -1’/C-1=VE/VC (similar triangles) =VF 1/VC=2/3 THEREFORE P 1 AND P 1’ LIE ON THE ELLIPSE (vertex) V 1 2 P 1’ P 2’ F 1 -P 2=F 1 -P 2’= 2 -2’ P 2 AND P 2’ ALSO LIE ON THE ELLIPSE B 9

ELLIPSE BY CONCENTRIC CIRCLE METHOD Draw ellipse by Concentric Circles method. Take major axis 100 mm and minor axis 70 mm long. 3 2 Steps: 1. Draw both axes as perpendicular bisectors of each other & name their ends as shown. 2. Taking their intersecting point as a center, draw two concentric circles considering both as respective diameters. 3. Divide both circles in 12 equal parts & name as shown. 4. From all points of outer circle draw vertical lines downwards and upwards respectively. 5. From all points of inner circle draw horizontal lines to intersect those vertical lines. 6. Mark all intersecting points properly as those are the points on ellipse. 7. Join all these points along with the ends of both axes in smooth possible curve. It is required ellipse. 4 C 1 3 2 (x, y) 5 4 b 1 5 A (0, 0) a B 10 10 6 9 8 D 9 7 6 7 8 10

Steps: 1 Draw a rectangle taking major and minor axes as sides. 2. In this rectangle draw both axes as perpendicular bisectors of each other. . 3. For construction, select upper left part of rectangle. Divide vertical small side and horizontal long side into same number of equal parts. ( here divided in four parts) 4. Name those as shown. . 5. Now join all vertical points 1, 2, 3, 4, to the upper end of minor axis. And all horizontal points i. e. 1, 2, 3, 4 to the lower end of minor axis. 6. Then extend C-1 line upto D-1 and mark that point. Similarly extend C-2, C-3, C-4 lines up to D-2, D-3, & D-4 lines. 7. Mark all these points properly and join all along with ends A and D in smooth possible curve. Do similar construction in right side part. along with lower half of the rectangle. Join all points in smooth curve. It is required ellipse. ELLIPSE BY RECTANGLE METHOD Draw an ellipse by Rectangle OR Oblong method. Take major axis 100 mm and minor axis 70 mm long. D 4 4 3 3 2 2 1 1 A 1 2 3 3 4 2 B 1 C 11

ELLIPSE Problem 3: Draw ellipse by Oblong method. Draw a parallelogram of 100 mm and 70 mm long sides with included angle of 750. Inscribe STEPSEllipse ARE SIMILAR in it. TO BY OBLONG METHOD THE PREVIOUS CASE (RECTANGLE METHOD) ONLY IN PLACE OF RECTANGLE, HERE IS A PARALLELOGRAM. D 4 4 3 3 2 2 1 1 A 1 2 3 C 4 3 2 1 B