# 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

### 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|, |C| approximately equal
5. A.B = 0
6. B.C = 0
7. C.A = 0

### 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. 0 < |H| < |A|