Transformations - Transformations and Projections in Computer Graphics - page 35

1.4.1 Reflection
It is easy to reflect a point ( x, y, z ) about any of the three coordinate planes xy , xz ,or
yz . All that's needed is to change the sign of one of the point's coordinates. In this
section, we discuss and explain the general case where an arbitrary plane and a point
are given and we want to reflect the point about the plane. We proceed in three steps
as follows: (1) We discuss planes and their equations, (2) show how to determine the
distance of a point from a given plane, and (3) explain how to compute the reflection
of a point about a plane.
The (implicit) equation of a straight line is Ax + By + C =0,where A or B but not
both can be zero. The equation of a flat plane is the direct extension Ax + By + Cz + D =
0, where A , B ,and C cannot all be zero. Four equations are needed to calculate the
four unknown coe cients A , B , C ,and D . On the other hand, we know that any three
independent (i.e., noncollinear) points P i =( x i ,y i ,z i ) ,i =1 , 2 , 3 define a plane. Thus,
we can write a set of four equations, three of which are based on three given points and
the fourth one expressing the condition that a general point ( x, y, z ) lies on the plane

xyz 1
x 1 y 1 z 1 1

0=

x 2 y 2 z 2 1

x 3 y 3 z 3 1

y 1 x 1 x 1 x 1

z 1 1 z 1 1 y 1 1 y 1 z 1

= x y 2 z 2 1 − y x 2 z 2 1 + z x 2 y 2 1 − x 2 y 2 z 2 .

y 3 z 3 1 x 3 z 3 1 x 3 y 3 1 x 3 y 3 z 3

We cannot solve this system of equations because x , y ,and z can have any values,
but we don't need to solve it! We just have to guarantee that this system has a solution.
In general, a system of linear algebraic equations has a solution if and only if its deter-
minant is zero. The expression below assumes this and also expands the determinant
by its top row:

xyz 1
x 1 y 1 z 1 1

0=

x 2 y 2 z 2 1

x 3 y 3 z 3 1

y 1 z 1 1 x 1 z 1 1 x 1 y 1 1 x 1 y 1 z 1

= x y 2 z 2 1 − y x 2 z 2 1 + z x 2 y 2 1 − x 2 y 2 z 2 .

y 3 z 3 1 x 3 z 3 1 x 3 y 3 1 x 3 y 3 z 3

This is of the form Ax + By + Cz + D = 0, so we conclude that

y 1 z 1 1 x 1 z 1 1 x 1 y 1 1 x 1 y 1 z 1

A = y 2 z 2 1 B = − x 2 z 2 1 C = x 2 y 2 1 D = − x 2 y 2 z 2 .

y 3 z 3 1 x 3 z 3 1 x 3 y 3 1 x 3 y 3 z 3

(1.24)