Use the barycentric coordinates with respect to S to determine if the point p is inside, outside, on a face, or on the edge
of conv S which is a tetrahedron.
-$$\mathrm { S } = \left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } , \mathbf { v } _ { 4 } \right\}$$
Barycentric coordinates: $\left( \frac { 1 } { 8 } , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \frac { 1 } { 8 } \right)$
A) Outside
B) Edge
C) Inside
D) Face

Question

Quizplus · Accepted Answer

Barycentric coordinates represent a point's position relative to a simplex (in this case, a tetrahedron) as a combination of the simplex's vertices. If all coordinates are positive and their sum is 1, the point is inside the simplex. Here, the coordinates sum to 1 ($\frac{1}{8} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = 1$) and all are positive, indicating the point is inside the tetrahedron.

Use the Barycentric Coordinates with Respect to S to Determine S={v1,v2,v3,v4}\mathrm { S } = \left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } , \mathbf { v } _ { 4 } \right\}S={v1​,v2​,v3​,v4​}

Use the Barycentric Coordinates with Respect to S to Determine $\mathrm { S } = \left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } , \mathbf { v } _ { 4 } \right\}$