Graphics Reference
In-Depth Information
v
=
d
APCD
/
d
ABCD
,
w
=
d
ABPD
/
d
ABCD
, and
x
=
d
ABCP
/
d
ABCD
=
1
−
u
−
v
−
w
of the following determinants:
p
x
b
x
c
x
d
x
p
y
b
y
c
y
d
y
p
z
b
z
c
z
d
z
1111
a
x
p
x
c
x
d
x
a
y
p
y
c
y
d
y
a
z
p
z
c
z
d
z
1111
a
x
b
x
p
x
d
x
a
y
b
y
p
y
d
y
a
z
b
z
p
z
d
z
1111
d
PBCD
=
,
d
APCD
=
,
d
ABPD
=
,
a
x
b
x
c
x
p
x
a
y
b
y
c
y
p
y
a
z
b
z
c
z
p
z
1111
a
x
b
x
c
x
d
x
a
y
b
y
c
y
d
y
a
z
b
z
c
z
d
z
1111
d
ABCP
=
,
and
d
ABCD
=
.
These determinants correspond to the signed volumes of the tetrahedra
PBCD
,
APCD
,
ABPD
,
ABCP
, and
ABCD
(strictly 1/6 of each signed volume). As shown further ahead,
the ratios simplify to being the normalized relative heights of the point over the
opposing planes.
Returning to triangles, just as the barycentric coordinates with respect to a tetrahe-
dron can be computed as ratios of volumes the barycentric coordinates with respect
to a triangle can be computed as ratios of areas. Specifically, the barycentric coor-
dinates of a given point
P
can be computed as the ratios of the triangle areas of
PBC
,
PCA
, and
PAB
with respect to the area of the entire triangle
ABC
. For this reason
barycentric coordinates are also called
areal coordinates
. By using signed triangle areas,
these expressions are valid for points outside the triangle as well. The barycentric
coordinates (
u
,
v
,
w
) are thus given by
u = SignedArea(PBC)/SignedArea(ABC),
v = SignedArea(PCA)/SignedArea(ABC), and
w = SignedArea(PAB)/SignedArea(ABC)=1-u-v.
Because constant factors cancel out, any function proportional to the triangle area
can be used in computing these ratios. In particular, the magnitude of the cross
product of two triangle edges can be used. The correct sign is maintained by taking
