Introduction

Bounding Objects

We are interested in quickly determining if two objects intersect. We gave some reasons why this was important last day.
If the two objects are complicated this might be computationally intensive.
One trick is to surround each object with a simple bounding object and to check for the intersection of the bounding objects. Only if the bounding objects intersect, do we do any more complicated intersection testing.
Definition. A bounding object for an object `A` is any object `B` that contains `A`.
We will typically be interested in bounding boxes:
Definition. A box in `\mathbb{R}^n` is any subset of the form
`[a_1, b_1]\times[a_2, b_2]\times \cdots\times [a_n, b_n],` where `a_i, b_i \in \mathbb{R}`. Here `[a_i, b_i]` are segments rather than intervals. i.e., we allow `b_i < a_i`.

It is easy to check the intersection of bounding boxes:
Theorem. The boxes
`X = [a_1, b_1]\times[a_2, b_2]\times \cdots\times [a_n, b_n]` and `Y = [c_1, d_1]\times[c_2, d_2]\times \cdots\times [c_n, d_n]`
will intersect iff
`lb_i = \max(\min(a_i, b_i), \min(c_i, d_i)) \leq ub_i = \min(\max(a_i, b_i), \max(c_i, d_i))`
for all `i`. If they intersect, then
`X \cap Y = [lb_1, ub_1]\times[lb_2, ub_2]\times \cdots\times [lb_n, ub_n]`.
Because we take min's and max's in the above calculation, the above test is called a minimax test.

One way to generalize the notion of a bounding box is to allow bounding faces to be slanted and not just horizontal or vertical.
Definition. A slab in `\mathbb{R}^n` is the region between two parallel hyperplanes.
Given a bounded set `X` (i.e., any set that lives in some finite radius ball in `\mathbb{R}^n`), a unit vector `\vec n` determines a slab as follows: Starting arbitrarily far out on the two ends of the line through the origin with direction vector `\vec n`, slide two hyperplanes orthogonal to `\vec n` towards the origin until they touch `X`.
We will call this slab the slab for X induced by the unit vector `\vec n` and denote it `Slab(X, \vec n)`.
The equations of the two hyperplanes will be `\vec n \cdot \vec p = d^(n\ear)` and `\vec n \cdot \vec p = d^(far)` where `d^(n\ear) \leq d^(far)`. We denote the former hyperplane the near hyperplane and the later the far hyperplane.
Definition. The generalized bounding box for `X` determined by a fixed set `\vec n_1,\vec n_2, \ldots, \vec n_k` of unit normal vectors is the intersection of the slabs they determine. i.e.,
`B(X, \vec n_1,\vec n_2, \ldots, \vec n_k) = \cap_{i=1}^k Slab(X, \vec n_i).`

For some types of objects finding a generalized bounding boxes is still straightforward:
- Bounded, Linear Polyhedra (a union of finitely many bounded, linear, convex polyhedra where a bounded, linear, convex polyhedra is a bounded, finite intersection of hyperplane halfspaces). In this case, we project all of the polyhedra's vertices `\vec p_j` onto the `\vec n_i` and use the minimum and maximum of those values. That is,
  `d_i^(n\ear) = \min_j(\vec n_i \cdot \vec p_j)` and `d_i^(far) = \max_j(\vec n_i \cdot \vec p_j)`
- Implicitly Defined Surfaces. Suppose `S` is defined by an equation `f(x,y,z) = 0`. The `d_i^(n\ear)` and `d_i^(far)` will be the minimum and maximum, respectively of the linear function `g(x,y,z) = \vec n_i\cdot(x,y,z)` subject to the constraint of being on the surface. This minimum and maximum can be found using Lagrange multipliers.
- Compound Objects. Assume that the object is defined by successive application of the regular set operations starting from primitive objects of the above two types. The `d_i^(n\ear)` and `d_i^(far)` can be computed in terms of the `d_i^(n\ear)` and `d_i^(far)` of the primitives that define it. For example, to get the `d_i^(n\ear)` and `d_i^(far)` of the union of two objects `X` and `Y`, one takes the minimum of the `d_i^(n\ear)`'s and the maximum of the `d_i^(far)`'s.

Suppose one has two generalized bounding boxes `B_1` and `B_2` each of which is defined with respect to the same set of normal vectors.
To determine if their intersect `B` we calculate minima of `d_i^(n\ear)`'s and maxima of the `d_i^(far)`'s:
`d_i^(n\ear)(B) = \max(d_i^(n\ear)(B_1), d_i^(n\ear)(B_2))` and `d_i^(far)(B) = \min(d_i^(far)(B_1), d_i^(far)(B_2))`
The generalized boxes `B_1` and `B_2` are disjoint iff `d_i^(far)(B) < d_i^(n\ear)(B)` for some `i`.

Another common task that we often need to do in geometric modeling is to determine if a point is inside a surface.
This task is called a surrounding test.
There are two main types of surrounding tests we will look at: those of convex polyhedra and those of arbitrary polyhedra.

We begin by considering the case of convex polyhedra.
We defined bounded, linear convex polyhedra earlier. A convex polyhedra (as a surface) `Q` is the intersection of a collection of halfplanes. These give the faces of `Q`. Each face `i` can be defined via a point `\vec q_i` on the face and a normal `\vec n_i`.
A point `\vec p` will belong to `Q` iff `(\vec p - \vec q_i)\cdot \vec n_i \geq 0` for all `i`.
This is called the normal test.

Sometimes it is more convenient to store planes, lines, etc. via equations of the form `a_1x_1 + a_2x_2 + \cdots + a_nx_n =0`, rather than use a normal and a point on the object.
If this is the case, then we can modify the normal test to get the so-called equation test.
In 2D, we notice `\vec n_i = (a_i, b_i)` and `c_i = - \vec q_i\cdot \vec n_i`.
So `\vec p = (x,y)` will belong to the polygon iff
`a_i x + b_iy +c_i \geq 0`
for all `i`.