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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

T

F

75100

75

Section plane

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

Axis

Draw horizontal circles around the conesurface with center coinciding with the axis

in the TV

Project corresponding points of intersection

of the circles with the section plane in FV to

the TV

Join these points to get the section

face

For true shape of the section, draw

an auxiliary view with reference line

parallel to the section planePRIMARY AUXILIARYVIEW

True shape of section (parabola)

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The plane is perpendicular to the top plane,

therefore the section line is drawn in the Top

View

It cuts the base at fandj

It cuts the edges at g and h

Join these points to o form the section face

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 of15 mm from the axis with section plane making an angle of50o with FP. Draw the

remaining part of the pyramid and the true shape of the cut section

50o

e

a

re

a b

co

cm

n

p

m

n

p100 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

Section

plane

50r

b

d

d

oT

F

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The pyramid is also cut by another plane that is perpendicular to the frontal plane, inclined at 70o

to the top plane and cuts the axis of the pyramid at 15mm from the apex. Draw the projections of

the remaining part of the pyramid and the true shape of the cut section

15

70o

Parallel

Axis of pyramid

ae b

o

c

a

d

c

b

e

g, i

h

h

h1

g1i1

jk

j1

k1

jk

100

g

i

Since the section plane is

perpendicular to the frontal plane, thesection line is drawn in the front view

The cutting plane cuts the axis of the

pyramid (light blue) 15 mm below the

apex

It cuts the base at g and i

It also cuts the edges at h, j,l and k

Project these points in the top view

and join them

Eliminate the edge oe and part of theedge oa which are cut off

Project an auxiliary view of the True

shape of the section by taking the

reference line parallel to the section

lineSection plane

l

l1

True length

l

d

T

F

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o

d

o

d

l

c

c

l

How to locate the point l

Draw an imaginary horizontal line from the axis

(light blue) to the edge oc intersecting at z

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 yand rotating.

Basically the imaginary line with length oz = oy

is rotating inside the pyramid from one edge to

another

This can also be obtained by drawing a line from

z in the Top view parallel to dc (as dc is TL here)

z

zy

y

b

b

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5

CONIC SECTIONSELLIPSE, 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

Ellipse

Section Plane Parallel

to end generator.

Section Plane

Parallel to Axis.Hyperbola

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6

These are the loci of points moving in a plane such that the ratio of its distances

from a f ixed pointAnd a fi xed linealways remains constant.

The Ratio is called ECCENTRICITY. (E)

A) For Ellipse E1

COMMON DEFINITION OF ELLIPSE, PARABOLA & HYPERBOLA:

PC

PFE

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Ellipse

7

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

12

2

2

2

b

y

a

x

Major axis

Minor axis

Focus

P

F1 F2

A B

C

D

PF1 + PF2 = constant

= AF1 + AF2 = AB =Length of Major axis

CF1 + CF2 = 2CF1= AB

CF1= AB/2

Found where?

Arches, bridges, dams, monuments,man-holes, glands

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Construct an ellipse, length of major and minor axis given:

Arcs of circles method

8

P

F1 F2

A B

C

D

1 2 3

Q R

PRINCIPLE: Sum of distances of point to foci = Length of major axis

F1P = A1, F2P = B1

F1Q = A2, F2Q = B2

F1R = A3, F2R = B3

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9

ELLIPSEDIRECTRIX-FOCUS METHOD

Draw an ellipse, focus is 50 mm from the directrix

and the eccentricity is 2/3

F1 (focus)V(vertex)

A

B

E

C

1

1

P1

P1

P2

P2

VE = VF1

F1-P1=F1-P1 = 1-1

2

2

F1-P2=F1-P2= 2-2

P2 AND P2 ALSO LIE ON THE ELLIPSE

F1-P1/(P1 to directrix AB) =1-1/C-1=VE/VC (similar triangles)

=VF1/VC=2/3

THEREFORE P1 AND P1 LIE ON THE ELLIPSE

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10

1

2

3

4

5

6

7

8

9

10

B

A

D

C

1

23

4

5

6

78

9

10

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 verticallines 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.

Draw ellipse by Concentric Circles method.

Take major axis 100 mm and minor axis 70 mm long.

ELLIPSEBY CONCENTRIC CIRCLE METHOD

(x,y)

(0,0)a

b

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11

1

2

3

4

1

2

3

4

A B

C

D

Draw an ellipse by Rectangle OR Oblong method.

Take major axis 100 mm and minor axis 70 mm long.

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 upperleft 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.

ELLIPSEBY RECTANGLE M ETHOD

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