Computer Aided Engineering Design

74 COMPUTER AIDED ENGINEERING DESIGN

to represent a circle in the first quadrant. Herein, the parametric form of representation becomes
useful. For two-dimensional curves, parametric equations can be written as

x = f(u)

y = g(u) (3.13)

whereu is the parameter. Note that the issue of vertical tangents easily gets resolved by using
f (u) = x 0 = constant and g(u) = u,u ranging from – ∞ to ∞. For two curves, [f 1 (u 1 ),g 1 (u 1 )] and
[f 2 (u 2 ),g 2 (u 2 )] to intersect, the equations

f 1 (u 1 ) = f 2 (u 2 ) and g 1 (u 1 ) = g 2 (u 2 ) (3.14)

can be solved for u 1 and u 2. Finally, curve segments can be represented by imposing the bounds on
the parameters. Thus, for a straight line segment between (x 1 ,y 1 ) and (x 2 ,y 2 )

x = (1 – u)x 1 + ux 2

y = (1 – u)y 1 + uy 2 0 ≤u≤ 1 (3.15a)

and for a circular arc of radius r between arguments θ 1 and θ 2

x = r cos θ

y=rsinθθ 1 ≤ θ≤θ 2 (3.15b)

With uas the parameter, the equation of a curve in three-dimensions can be written in compact vector
form as

r(u) = x(u)i + y(u)j + z(u)k (3.16)

wherex(u),y(u) and z(u) are scalar functions of u. Many analytic curves may be represented in the
above parametric form. For instance, the equation of a circle of radius a in terms of parameter u = ωt
is given by

r(t) = a cos(ωt)i + a sin(ωt)j (3.17a)

where a particle may be considered traversing on the circumference with an angular velocity ω at
timet. Similarly, parametric equations for an ellipse, parabola, hyperbola and cylindrical helix can be
expressed, respectively, by

r(u) = a cos(u)i + b sin(u)j

r(u) =u^2 i + 2 a1/2uj

r(u) = a sec(u)i + b tan(u)j

r(u) = a cos(u)i + a sin(u)j + buk (3.17b)

Curves of intersection between solids like cylinders, cones and spheres are often encountered in
engineering design. One such example is the intersection curve between a cylinder [(x– a)^2 + y^2 =a^2 ]
and a sphere [x^2 + y^2 + z^2 = 4 a^2 ] known as Viviani’s curve (Figure 3.5) whose parametric equation
may be written as

r(u) = a(1 + cosu)i+a sin uj+ 2a sin^1

2

uk (3.17c)