1. ENGINEERING GRAPHICS
Conic sections: ellipse, parabola, hyperbola.
Special curves: cycloid, epicycloid, hypocycloid
and involutes.
Department of Mechanical Engineering
Dr. G.Rajesh
3. Conics defined: It is a locus of point which moves in a
plane, in such a way that the ratio of its distance from the focus
to its distance from the directrix is always a constant
Terminology:
The fixed point is called the Focus
The fixed line is called the Directrix
Axis is the line passing though the
focus and perpendicular to the
directrix
Vertex is a point at which the conic
cuts its axis
5. Ellipse defined: It is a plane curve generated by a point which
moves in such a way that, at any position the sum of the distance
from two fixed points is always a constant. The constant equal to
the major axis of the ellipse.
Ellipse (e < 1)
Eccentricity
e = Distance of moving point from focus
Distance of moving point from directrix
7. Eccentricity method
1. Draw the locus of a point P moving so that the ratio of the distance
from focus F to its directrix DD’ is 2/3 . Distance of focus from directrix
is 50 mm. Also draw tangent & normal at a point 40 mm from directrix.
9. Parabola (e = 1)
Applications
Heater, Microphone, Dish, suspension bridges
It is a plane curve generated by a point which moves in such a
way that, at any position its distance from a fixed point (focus) is
always equal to its distance from a fixed straight line (directrix).
10. Hyperbola (e > 1)
Applications
Telescope optics, Natural draft cooling towers
It is a plane curve generated by a point which moves in such a
way that, at any position the difference of its distance from two
fixed points is always a constant.
11. 2. Draw the path traced by a point P moving in such a way that the
distance of the focus from directrix is 40 mm. The eccentricity is unity.
(or)
Construct a parabola when the distance between the focus and vertex is
20 mm. Draw the tangent and normal to a point on the parabola 40 mm
from the focus.
Parabola
12. 3. Trace and name the locus of a moving point P, having the ratio of eccentricity 4/3
and the distance between the focus to the vertex is 40 mm. Also draw a tangent and
normal at any point on the curve.
(Or)
The vertex of the hyperbola is 20 mm from the directrix. Draw the curve If the
eccentricity is 3/2. Also draw the tangent and normal to a point on the curve 50 mm
from the axis.
Hyperbola
13. Cycloidal Curves
A cycloid is a curve generated by point on the circumference of a circle which
rolls in a plane surface along a straight line without slipping. The rolling circle
is called a generating circle. The straight line is called as base line.
Cycloid
14. A epicycloid is a curve generated by point on the circumference of a circle
which rolls in a plane along outside of another circle. The rolling circle is
called a generating circle. The circle on which the generating circle rolls is
called as directing circle or base circle.
Epicycloid
15. Hypocycloid
A hypocycloid is a curve generated by point on the circumference of a circle
which rolls in a plane along inside of another circle.
17. 4.
4. A cycle wheel of 1 m diameter rolls on a straight line without slipping.
Trace the locus of point P on the circumference of the wheel which is
rolling for one complete revolution. Draw tangent and normal at any
point on the curve. (assume suitable scale)
18. 5. Construct an epicycloid generated by a circle of diameter 50 mm
when it rolls over an another circle of diameter 150 mm . Draw the
tangent and normal at any point on the epicycloid.
19. 6. Construct a hypocycloid generated by a rolling circle of diameter 50
mm which rolls inside a base circle of diameter 150 mm. Draw tangent
and normal at any point on the curve.
20. Involute
An involute is a spiral curve generated by a point on a cord or
thread as it unwinds from a polygon or a circle, as the thread
being kept tight.
Applications
Gear tooth profiles, Pump casing , Gas compressor (scroll compressor)
21. 7. Draw the curve traced out by an end of a string unwound from a
regular pentagon of side 20 mm, the string being kept taut. Draw a
tangent and normal to the curve at a point 100 mm from the centre of
the pentagon
22. 8. Construct one convolution of an involute of a hexagon of side 25 mm.
23. 9. Construct one convolution of an involute of a circle of diameter 30
mm. Draw tangent and normal at a point on the involute 65mm from
the centre of the circle.
