A Survey of Implicit Constraints in Primitives

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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

1. 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

1. Modulus Comparison : Comparison of the modulus of a vector to a real number ( 0 for non-negativity ) or the modulus of another vector
2. 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:

1. |A| > 0
2. |B| > 0
3. |C| > 0
4. A.B = 0
5. B.C = 0
6. C.A = 0

rec (Right elliptical cylinder)

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

Constraints:

1. |H| > 0
2. |A| > 0
3. |B| > 0
4. A = C
5. B = D
6. A.B = 0
7. H.A = 0
8. H.B = 0

rhc (Right hyperbolic cylinder)

3 types: Non-negativity/Modulus comparison, Perpendicularity

Constraints:

1. |H| > 0
2. |B| > 0
3. |R| > 0
4. H • B = 0
5. c > 0
6. |B| ≥ c

rpc (Right parabolic cylinder)

2 types: Non-negativity/Modulus comparison, Perpendicularity

Constraints:

1. |H| > 0
2. |B| > 0
3. |R| > 0
4. H.B = 0

sph (Sphere)

Sphere is a particular case of the ellipse

Constraints: 2 types: Modulus comparison, Perpendicularity

1. |A| > 0
2. |B| > 0
3. |C| > 0
4. |A| = |B|
5. |A| = |C|
6. |B| = |C|
7. A.B = 0
8. B.C = 0
9. C.A = 0

tgc (Truncated General Cone)

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

1. |H| > 0
2. |A| & |B| not zero together
3. |B| & |D| not zero togehter
4. |A|*|B| and |C|*|D| not zero together
5. H is nonplanar to AB plane
6. A.B = 0
7. C.D = 0
8. 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

1. |A| = |B|
2. A.B = 0
3. B.H = 0
4. H.A = 0
5. |H| > 0
6. |H| < |A|