Game Development Reference
In-Depth Information
Converting a normal and a point on the line to normal and distance
form:
distance =
n
q
.
(This assumes that
n
is a unit vector.)
Finally, to convert perpendicular bisector form to implicit form, we use
a = q
y
− r
y
,
b = r
x
− q
x
,
d =
q
+
r
2
=
q
+
r
2
a b
q
y
− r
y
r
x
− q
x
=
(q
x
+ r
x
)(q
y
− r
y
) + (q
y
+ r
y
)(r
x
− q
x
)
2
=
(q
x
q
y
− q
x
r
y
+ r
x
q
y
− r
x
r
y
) + (q
y
r
x
− q
y
q
x
+ r
y
r
x
− r
y
q
x
)
2
= r
x
q
y
− q
x
r
y
.
9.3
Spheres and Circles
A sphere is a 3D object defined as the set of all points that are a fixed
distance from a given point. The distance from the center of the sphere to
a point is known as the radius of the sphere. The straightforward repre-
sentation of a sphere is to describe its center
c
and radius r.
Spheres appear often in computational geometry and graphics because
of their simplicity. A bounding sphere is often used for trivial rejection
because the equations for intersection with a sphere are simple. Also im-
portant is that rotating a sphere does not change its extents. Thus, when
a bounding sphere is used for trivial rejection, if the center of the sphere is
the origin of the object, then the orientation of the object can be ignored.
A bounding box (see
Section 9.4)
does not have this property.
The implicit form of a sphere comes directly from its definition: the set
of all points that are a given distance from the center. The implicit form
of a sphere with center
c
and radius r is
Implicit definition of a
sphere using vector
notation
p
−
c
= r,
(9.7)
where
p
is any point on the surface of the sphere. For a point
p
inside the
sphere to satisfy the equation, we must change the “=” to a “≤”. Since
Equation (9.7) uses vector notation, it also works in 2D, as the implicit
definition of a circle. Another more common form is to expand the vector
notation and square both sides:
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