Vectors - Essentials of Interactive Computer Graphics - page 479

B.3 Vector Examples

This section contains a number of examples of using vectors for geometric com-
putations. These map fairly directly into code when you use a library that has
functions or operators for things such as vector addition.

B.3.1 Implicit Curves in 2D

An implicit curve in 2D is a real-valued function f ( x , y ) that divides the plane into

positive and negative regions. The boundary where f ( x , y )= 0 is the curve. The

reason it is called “implicit” is that f can tell you whether a given point is on

the curve, but it does not tell you explicitly which points are on the curve. For
example, the circle centered at the origin of radius 1 is often written

()

x 2 y 2 + = 1 .

That is not quite in the form f ( x , y )= 0. We can subtract 1 from both sides to get

x 2 y 2 + − 1 = 0 .

For points inside the circle, x 2 y 2 0. For points outside the circle, x 2

+ − 1 < +

y 2 − 0.

Note that for any implicit function f

1 >

0, the curve is unchanged if we
multiply it by a non-zero constant. For example, a unit circle is also described by

( x , y )=

2 x 2 2 y 2 + − 2 = 0

and

2 x 2 2 y 2 − − + 2 = 0 .

The constant can change how positive or negative points off the circle are, and in
the case of a negative constant, it reverses which regions off the curve are positive
versus those that are negative. However, points on the curve remain zero.
A more general circle with radius R and center C is given by:

2 2

R 2 f ( x , y )=( x − x c ) +( y − y c ) − = 0

The same principles of implicit equations apply: points where f ( x , y )= 0areon

the curve, and points where f 0 are not on the curve. Both in math and in

code, vectors give a more compact and less error-prone way to describe a circle:

( x , y ) =

R 2 ( P − C ) · ( P − C ) − = 0 .