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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