Graphics Reference
In-Depth Information
P is on the plane is
(
P
C
) ·
N
=
0
.
Colinear points.
Points lying
If the plane has three noncolinear points C , D ,and E , the normal vector is
on the same line.
N
.
There are no implicit curves in 3D because a curve cannot divide 3D space
into separate regions like a surface can. However, parametric equations work
well for curves in 3D. “Mileage markers” can be places along a road that curve in
x , y ,and z . The parametric equation for a 3D line is the same as for a 2D line:
=(
D
C
) × (
E
C
)
P
(
s
)=
C
+
s
(
D
C
) .
Again, as the real number s varies from 0 to 1, the point P varies from C to D .If
s can take on any value, you get the whole line.
B.3.4
Intersection of a Line and a Circle/Sphere
Recall that in vector form, the implicit 2D circle and 3D sphere are the same:
R 2
(
P
C
) · (
P
C
)
=
0
.
Here, C and R are fixed, and P can be thought of as an argument we are testing.
In both 2D and 3D, the parametric equation for a line through points A and B is
P
(
s
)=
A
+
s
(
B
A
) .
This can be read, “given s , here is the corresponding point P on the line." For
any point P on the line, there is a unique s and vice versa (an invertible unique
mapping like that is called a bijection). This line and circle/sphere intersect if
there is some P
(
s
)
where the P satisfies the implicit equation for the circle/sphere,
that is,
˙
R 2
(
P
(
s
)
C
)
(
P
(
s
)
C
)
=
0
.
Expanding P
(
s
)
gives
˙
R 2
(
A
+
s
(
B
A
)
C
)
(
A
+
s
(
B
A
)
C
)
=
0
.
Because the dot product is distributive, we can separate the terms as a polynomial
in s :
+ (
R 2 .
s 2
[(
B
A
) · (
B
A
)]
+
2
[(
B
A
) · (
A
C
)]
s
A
C
) · (
A
C
)
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