Physically Based Animation - Computer Animation: Algorithms and Techniques - page 245

1. The parametric position function, P ( u , v )
2. The surface normal function, N ( u , v )
3. The implicit function, I ( x )

The parametric position function and surface normal function are expressed as functions of surface
parameters u and v (although for a given model, the surface parameters may be more complex, incor-
porating identifiers of object parts, for example). Given a value for u and v , the position of that point in
space and the normal to the surface at that point are given by the functions. The implicit function takes
as its parameter a position in space and evaluates to an approximation of the signed distance to
the surface where a point on the surface returns a zero, a point outside the object returns a positive
distance to the closest point on the surface, and a point inside the object returns a negative distance
to the closest point on the surface. These functions will, of course, also be functions of the current state
of the model,

c .

Useful constraints
The functions above can be used to define several useful constraint functions. The methods of Witkin,
Fleischer, and Barr [ 26 ] can also lead to several intuitive constraint functions. The functions typically
are associated with one or more user-specified weights; these are not shown.

Point-to-fixed-point
The point Q in space is fixed and is given in terms of absolute coordinate values. The point P is a spe-
cific point on the surface of the model and is specified by the u , v parameters. The energy function will
be zero when these two points coincide.

2

E ¼j P ðu; vÞ Q j

Point-to-point
A point on the surface of one object is given and the point on the surface of a second object is given. The
energy function will be zero when they coincide. Notice that within a constant, this is a zero-length
spring. Also notice that the orientations of the objects are left unspecified. If this is the only constraint
given, then the objects could completely overlap or partially penetrate each other to satisfy this
constraint.

P a P b

2

E ¼j ðu a ; v a Þ ðu b ; v b Þj

Point-to-point locally abutting
The energy function is zero when the points coincide, and the dot product of the normals at those points
is equal to 1 (i.e., they are pointing away from each other).

2

ðu a ; v a Þ

P a P b N a N b

E ¼j ðu a ; v a Þ ðu b ; v b Þj þ ðu b ; v b Þþ :

1 0

Floating attachment
With the use of the implicit function of object b , a specific point on object a is made to lie on the surface
of object b .

I b P a

2

E ¼ð ð ðu a ; v a ÞÞÞ