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

1
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

T

• Draw horizontal circles around the cone surface
  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

F
100
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 plane
75
Axis
2
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 50o with FP. Draw the
remaining part of the pyramid and the true shape
of the cut section
d
The plane is perpendicular to the top plane,
therefore the section line is drawn in the Top
View It cuts the base at f and j It cuts the
edges at g and h Join these points to o form the
section face
Section plane
15
r
e
c
p
o
n
m
T
a
b
50o
o
F
p1
n1
r1
n
p
100
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
m1
m
e
b
c
r
a
d
50
3
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
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 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
the edge 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 line
True length
i
l
c
e
g
k
Parallel
j
h
b
T
a
70o
k1
F
o
l1
j1
k
15
j
l
h
h1
100
i1
g, i
g1
e
c
b
d
a
Axis of pyramid
Section plane
4
d
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
y and 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)
l
c
o
z
y
b
o
l
z
y
c
d
b
5
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.
Ellipse
Section Plane Through Generators
Section Plane Parallel to Axis.
Hyperbola
Parabola
Section Plane Parallel to end generator.
6

• These are the loci of points moving in a plane
  such that the ratio of its distances
• from a fixed point And a fixed line always
  remains constant.
• The Ratio is called ECCENTRICITY. (E)
• For Ellipse Elt1
• For Parabola E1
• For Hyperbola Egt1

COMMON DEFINITION OF ELLIPSE, PARABOLA
HYPERBOLA
7
Ellipse
Equation a Half length of major axis b Half
length of minor axis Eccentricity e lt 1 Sum of
distances of a point on the ellipse to the foci
is constant
Found where? Arches, bridges, dams, monuments,
man-holes, glands
C
Minor axis
P
PF1 PF2 constant AF1 AF2 AB Length of
Major axis
Major axis
A
B
F1
F2
CF1 CF2 2CF1 AB CF1 AB/2
Focus
D
8
Construct an ellipse, length of major and minor
axis given Arcs of circles method
C
Q
R
P
1
2
3
A
B
F1
F2
D
PRINCIPLE Sum of distances of point to foci
Length of major axis F1P A1, F2P B1 F1Q A2,
F2Q B2 F1R A3, F2R B3
9
ELLIPSE DIRECTRIX-FOCUS METHOD
A
VE VF1
2
1
DIRECTRIX
F1-P1F1-P1 1-1
P2
E
P1
F1-P1/(P1 to directrix AB) 1-1/C-1VE/VC
(similar triangles) VF1/VC2/3 THEREFORE P1 AND
P1 LIE ON THE ELLIPSE
V
F1 ( focus)
(vertex)
C
1
2
P1
F1-P2F1-P2 2-2 P2 AND P2 ALSO LIE ON THE
ELLIPSE
P2
B
10
ELLIPSE BY CONCENTRIC CIRCLE METHOD
11
ELLIPSE BY RECTANGLE METHOD
4

3
2
1
12
ELLIPSE BY OBLONG METHOD
4
3
2
1