What are barycentric coordinates?

Barycentric coordinates are a mathematical tool used to represent points in a geometric space. It describes points with the vertices of a geometric object, like a triangle or tetrahedron. It makes them useful in computer graphics and computational geometry, where working with triangles is common. 

Why barycentric coordinates?

Barycentric coordinates play an essential role in many aspects of computer graphics and 3D modeling, mainly due to the significant use of triangles. Given below are the major reasons why barycentric coordinates are used.

• Interpolation across triangles: Barycentric coordinates allow for smooth interpolation across the surface of a triangle. It is used when we want to interpolate or extrapolate the values from the vertices to any point within the triangle. These values can be texture coordinates, color, or other data associated with the vertices. Using barycentric coordinates makes the interpolation process efficient and straightforward.

• Simplex representation: Triangles are the basic building blocks of 3D modeling. They are stored as a sequence of three vectors representing a vertex. Barycentric coordinates provide a way to work directly with these vertices, leading to a more straightforward understanding of the triangle and the associated points.

• Texture mapping: Barycentric coordinates play an important role in texture mappingThe process of applying a 2D image on a 3D model.. For every point within a triangle on the 3D model, the corresponding point on the 2D texture can be determined using the same barycentric coordinates. It ensures proper texture mapping from the 2D image onto the surface of the 3D model.

Cartesian coordinates vs. Barycentric coordinates

1. Cartesian coordinates

Cartesian coordinates are a way of representing points in space using distances along each dimension from a defined origin. In 2D, a point is represented by an $(x, y)$ pair, while in 3D, it is represented by an $(x, y, z)$. Each coordinate value tells how far the point is from the origin in its respective dimension.

Consider a 2D Cartesian coordinate system with an origin at point $O(0, 0)$:

• Point B (-1, 4): It means that the point is 1 unit to the left (negative x-direction) and 4 units up (positive y-direction) from the origin.

In 3D, we add a $z$-coordinate, and the points are represented by $(x, y, z)$.

2. Barycentric coordinates

Barycentric coordinates offer an alternative way of representing points within the geometric figures, most commonly used for triangles in 2D and 3D. Instead of using absolute distances, barycentric coordinates express a point as a combination of the vertices of the figure, with weights that add up to 1.

For a 2D triangle with vertices $a$, $b$, and $c$, and a point $P$ inside the triangle, the barycentric coordinates are represented as $(u, v, w)$.

How to calculate barycentric coordinates

Assume we have a triangle with vertices A, B, and C. We want to find the barycentric coordinates of a point P.

Barycentric coordinates are a form of a coordinate representation where each point within a triangle is represented as a weighted sum of the vertices. If we denote these weights as $\alpha$, $\beta$, and $\gamma$ corresponding to the vertices $a$, $b$, and $c$, then any point $P$ coplanar to the triangle can be represented as given below.

These weights have specific properties in the context of a barycentric coordinates.

1. $\alpha$, $\beta$, $\gamma$$\ge 0$

2. $\alpha$$+$ $\beta$$+$ $\gamma$$\; \; = 1$

Note:

Any point $p$ is inside the triangle if and only if:

• $0 < \alpha < 1$

• $0 < \beta < 1$

• $0 < \gamma < 1$

• If one of the weight component is zero, $P$ is on the edge.

• If two weight components are zero, $P$ is on the vertex.

Calculating barycentric coordinates using area ratios

• Take a point $P$ on the triangle

• Determine the coordinates by solving the system of linear equations

We can rearrange the above equation to the one shown below:

We can represent the above equation in matrix form, as shown below:

We can find the area of the triangle $abc$ by using the formula mentioned below.

And we can compute the barycentric coordinates as follows.

Note:

If a point $P$is closer to the vertex, the area of the triangle formed by that vertex will be larger.

The $\alpha$, $\beta$, and $\alpha$are the barycentric coordinates for the point $p$relative to the triangle $abc$.

Conclusion

In conclusion, barycentric coordinates enable us to describe the position of a point within a simplex (like a triangle or tetrahedron) in a highly intuitive way. Although understanding and calculating barycentric coordinates might seem difficult at first, with a solid grasp of the underlying mathematics and a little practice, they become a valuable tool in computational geometry.

Continue reading

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