Now that we have some 3D primitives defined, it's time to implement some simple point tests for them. In this chapter, we are going to implement the following point-related test functions:
Point contained in sphere
Closest point on sphere
Point contained in AABB
Closest point on AABB
Point contained in OBB
Closest point on OBB
Point on surface of plane
Closest point on plane
Point on line segment
Closest point along line
Point on ray
Closest point along ray
Given a point and a sphere there are two operations we want to perform. First, we want to check whether a test point is inside the sphere or not. Alternately, we may want to get the closest point to the test point along the surface of the sphere:
To test whether a point is within a sphere we have to compare the distance from the center of the sphere and the test point to the radius of the sphere. If the distance is less than the radius, the sphere contains the point. We can get the point on the surface of the sphere closest to a test point by obtaining a vector that points from the center of the sphere to the test point. This vector should have the same magnitude as the radius of the sphere.
If we think of an Axis Aligned Bounding Box (AABB) as a Min and Max point, a test point is only inside the AABB if it is greater than Min and less than Max. Similarly, to get the closest point to a test point on the surface of the AABB, we just have to clamp the test point to the min and max points of the AABB:
We are going to implement a function to test if a point is contained within an Axis Aligned Bounding Box. This test will compare the point component-wise to the min and max points of the AABB. We are also going to implement a function to find the point on the Axis Aligned Bounding Box closest to a given test point. To find the closest point, we will clamp the test point to the min and max points of the AABB, component-wise.
To test if a point is inside an Oriented Bounding Box (OBB), we could transform the point into the local space of the OBB, and then perform an AABB containment test. However, transforming the point into the local space of the OBB is needlessly expensive.
A more efficient solution is to project the point onto each axis of the OBB, then compare the projected point to the length of the OBB on each axis. To get the closest point to a test point on the surface of the OBB, we perform the same projection. Once the point is projected, we clamp it to the length of the OBB on each axis:
This diagram demonstrates a test point being projected and clamped to the Y axis of the OBB.
The test point must also be projected and clamped to the X and Z axes as well.
We have seen the plane equation before; a point is on a plane if the result of the plane equation is 0. To find the point on the plane closest to a test point, we must project the test point onto the normal of the plane. We then subtract this new vector from the test point to get the closest point:
We are going to implement two functions. The first function will test whether a point is on the surface of a plane using the plane equation. The second function will find the point on a plane closest to a given test point.
Perform the following steps to implement point tests for a plane:
Declare PointOnPlane and ClosestPoint in Geometry3D.h:
bool PointOnPlane(const Point& point, const Plane& plane); Point ClosestPoint(const Plane& plane, const Point& point);
Implement PointOnPlane in Geometry3D.cpp:
bool PointOnPlane(const Point& point, const Plane& plane) {
float dot = Dot(point, plane.normal);
// To make this more robust, use...To test if a point is on a line, or to get the point on a line closest to a test point, we first have to project the point onto the line. This projection will result in a floating point value, t. We use this new t value to find the distance of the point along the line segment using the distance(t) = start + t * (end - start)function. The start point of the line is at t = 0, the end point is at t = 1. We have to take two edge cases into account, when t is less than 0 or greater than 1:
We are going to implement two functions, one to get the point on a line closest to a test point and one to determine if a test point is on a line. The ClosestPoint function is going to project the test point onto the line and evaluate the parametric function, distance(t) = start + t * (end - start).
To determine if a test point is on a line segment, we still need the point on the segment closest to the test point. We are then able to measure the distance between the test point and...
A ray is the same as a directed line. Unlike a line segment, which has a start and an end point, a ray has only a start point and a direction. The ray extends infinitely in this one direction. Because of the ray's similarity to a line, operations on a ray are similar to those on a line.
Because a ray's direction is a normal vector, we can use the dot product to check its direction against other known vectors. For example, to test whether a point is on a ray, we need to get a normalized vector from the origin of the ray to the test point. We can then use the dot product to see if this new normal vector is the same as the normal of the ray. If two vectors point in the same direction, the result of the dot product will be 1:
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