A Survey of Implicit Constraints in Primitives

Types of Implicit Parameters

At the level of constraint networks, calculations are done in terms of Variables or indpendent real values / floating point numbers. But in the construction of geometry these are clustered together in terms of implicit parameters. Typical implicit parameters are

Vectors - A 3 dimensional vector is a 3-tuple which is used to hold direction as well as magnitude. In BRL-CAD primitives, they may represent
1. Radius vectors ( Center of a sphere)
2. Direction vectors (Direction of a plane)

Types of Implicit Constraints

An enumeration of the set of contraints observed in the primitives below

Modulus Comparison : Comparison of the modulus of a vector to a real number ( 0 for non-negativity ) or the modulus of another vector
Perpendicularity of Vectors

Implict Constraints by Primitive

ell (Ellipse)

Ellipse is built using the Center (radius vector V) and 3 Vectors (A, B, C st. |A| = radius) 2 types: Non-negativity/Modulus comparison, Perpendicularity Constraints:

|A| > 0
|B| > 0
|C| > 0
A.B = 0
B.C = 0
C.A = 0

rec (Right elliptical cylinder)

3 types: Non-negativity/Modulus comparison, Perpendicularity, Vector equality

Constraints:

|H| > 0
|A| > 0
|B| > 0
A = C
B = D
A.B = 0
H.A = 0
H.B = 0

rhc (Right hyperbolic cylinder)

3 types: Non-negativity/Modulus comparison, Perpendicularity

Constraints:

|H| > 0
|B| > 0
|R| > 0
H • B = 0
c > 0
|B| ≥ c

rpc (Right parabolic cylinder)

2 types: Non-negativity/Modulus comparison, Perpendicularity

Constraints:

|H| > 0
|B| > 0
|R| > 0
H.B = 0

sph (Sphere)

Sphere is a particular case of the ellipse

Constraints: 2 types: Modulus comparison, Perpendicularity

|A| > 0
|B| > 0
|C| > 0
|A| = |B|
|A| = |C|
|B| = |C|
A.B = 0
B.C = 0
C.A = 0

tgc (Truncated General Cone)

Constraints: 5 types: Modulus comparison, Logical Combination, Perpendicularity, Non-planarity, Parallelism

|H| > 0
|A| & |B| not zero together
|B| & |D| not zero togehter
|A||B| and |C||D| not zero together
H is nonplanar to AB plane
A.B = 0
C.D = 0
A || C ( A is parallel to C )

tor (Torus)

Tor is built using the following input fields

V    V from origin to center
H    Radius Vector, Normal to plane of torus.  |H| = R2
A, B     perpindicular, to CENTER of torus.  |A|==|B|==R1
F5, F6   perpindicular, for inner edge (unused)
F7, F8   perpindicular, for outer edge (unused)

Constraints: 2 types: Modulus comparison, Perpendicularity

|A| = |B|
A.B = 0
B.H = 0
H.A = 0
|H| > 0
|H| < |A|