Computer Aided Engineering Design

COMPUTATIONS FOR GEOMETRIC DESIGN 293

The above expression is in the implicit form in both s and t. For some values of s and t, let g 1 (s,t)
andg 2 (s,t) both not be equal to zero. Define G(s,t) = [g 1 (s,t)g 2 (s,t)]T and consider its first order
linear expansion, that is

Gs st t Gst

g

s

g

t

g

s

g

t

s

t

( + , + ) = ( , ) + =

0

0

1 1

22

ΔΔ

Δ

Δ

∂

∂

∂

∂

∂

∂

∂

∂

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

⎤

⎦

⎥

The intent in the above expression is that for some (iterative) revision (Δs,Δt) in the values of (s,
t),G(s,t) becomes 0. Rearranging above yields

∂

∂

∂

∂

∂

∂

∂

∂

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

⎤

⎦

⎥

∂

∂

∂

∂

∂

∂

∂

∂

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

g

s

g

t

g

s

g

s

s

t

gst

gst

s

t

g

s

g

t

g

s

g

s

1 1

22

1

2

1 1

22

= –

(, )

(, )

or = –

Δ

Δ

Δ

Δ

–1–1

1

2

(, )

(, )

gst

gst

⎡

⎣

⎢

⎤

⎦

⎥

where

∂

∂

∂

∂

⎡

⎣

⎢

⎤

⎦

⎥

∂

∂

⎡

⎣⎢

⎤

⎦⎥

∂

∂

⎡

⎣⎢

⎤

⎦⎥

gst

s

ts

s

s

s

s

s

s

T T

1 2

2

(, )

= [ ( ) – ( )]

()

• 
  

() ()

bc

ccc

∂

∂

∂

∂

⎡

⎣⎢

⎤

⎦⎥

∂

∂

⎡

⎣⎢

⎤

⎦⎥

∂

∂

gst

t

t

t

s

s

gst

s

T

12 (, ) = bc() ( ) = – (, )

∂

∂

∂

∂

⎡

⎣

⎢

⎤

⎦

⎥

∂

∂

⎡

⎣⎢

⎤

⎦⎥

∂

∂

⎡

⎣⎢

⎤

⎦⎥

gst

t

ts

t

t

t

t

t

t

T T

2 2

2

(, )

= [ ( ) – ( )]

()

+

() ()

bc

bbb

Thus starting with the initial values of s and t,Δs and Δt can be computed using the above
expressions. Parameters can be updated as s = s + Δs and t = t + Δt and using these new values,
G(s,t) = [g 1 (s,st)g 2 (s,t)]Tcan be computed. The procedure can be iterated until G(s,t) is desirably
close to 0. Note here that s and t values should not be allowed to assume values outside the interval
[0, 1]. An intersection point is obtained when d^2 = g(s,t) is adequately close to zero.

Exercises

1. Given a line A + td and a plane with base point B and normal vector n, what is the condition for the line
   to be perpendicular to the plane? What is the condition for the line to be parallel to the plane?


2. Find the proximity of the points (0, 0), (1, 5) and (1, 0) with respect to the line whose end points are A
   (1, 1) and B (1, 8).


3. Consider the line segments whose end points are AB (0, 0) (5, 0); BC (5, 0) (5, 5); CD (5, 5) (0, 5) and
   DA (0, 5) (0, 0). Find the positioning of the point P (1, 1) with respect to these lines. Comment on the
   membership (inside/outside/on) of P in polygon ABCD.


4. A quadrilateral is represented by the vertices A (2, –2), B (0, 15), C (–2, –2) and D (0, 4). Determine if
   pointE (0, –2) lies within this polygon. (Hint: The ray passes through the vertex A, thus infinitesimally
   shift the y coordinate of the ray and then perform the crossings test).