A way to think about why the alpha, beta, and gamma add to 1 is that they essentially represent each vertex's contribution in terms of "pulling" the point toward itself (1 being having pulled the point all the way to its location). If each vertex pulls on it equally (alpha = beta = gamma = 1/3), then the point sits in the middle of the triangle.
lycorisradiatu
@ danielhsu021202 By "pulling," do you mean the distance between the point and the vertices? Basically the shorter the distance, the stronger the "pulling"? That is a really interesting way to think of it. I really appreciate your sharing.
Songbird94
I’m wondering how does it work intuitively if outside the triangle
agao25
@Songbird94 I think if a point is outside the triangle, then at least one of alpha, beta, or gamma is going to be negative. And intuitively, it's like the contribution of vertices A, B, and/or C is "pulling" the point outside the triangle. There's probably a bunch of other tests that the programs we write will use to evaluate if a point is inside or outside a triangle
elaineqian02
Barycentric coordinates are especially useful for showing triangle geometry properties that do not depend on the angles - including Ceva's theorem, Menelaus's theorem, Routh's theorem, and Stewart’s theorem. Another cool application is in solving math olympiad geometry problems - this paper explains some of the theory and example problems: https://web.evanchen.cc/handouts/bary/bary-full.pdf
spegeerino
It's kind of interesting how you could use this to prove that 3 points define a plane; we can specify any point on the plane as a linear combination of the three points of this triangle (with the sum of the coefficients always being fixed at 1). Even though from a linear algebra perspective, the plane is 2-dimensional, it still takes 3 points to define the plane which means we have one dimension of freedom, which in this case is the sum of the coefficients. We could also fix the sum of the coefficients as 0, or -1, or pi, or whatever other number we want and still get a coordinate system, albeit one that's less relevant to the original triangle we were working with. One semi-interesting thing is that if we fix the sum as c, we can recover the barycentric coordinates of a copy of the triangle scaled by c by multiplying each point by c and restoring the sum of the coefficients to be 1.
saif-m17
Barycentric coordinates have an interesting connection to convexity, since all points in the triangle are a convex combination of the vertices (i.e. all alpha, beta and gamma must be positive for the point to lie in the triangle). Since their sum is bounded by one, if all are positive, then they will all be between 0 and 1, which satisfies the definition of a convex combination. If any of these weights is negative, the point will lie outside of the triangle.
AlsonC
Super interesting, reminds me of how I used the shoelace theorem a lot to complete geometry math problems
zy5476
Barycentric Coordinates feel like a very interesting and useful way of defining points on a triangle and seem to have many useful properties. I'm interested in knowing what other use cases they have as a quick search seemed to reveal that they were mostly used only within computer graphics.
A way to think about why the alpha, beta, and gamma add to 1 is that they essentially represent each vertex's contribution in terms of "pulling" the point toward itself (1 being having pulled the point all the way to its location). If each vertex pulls on it equally (alpha = beta = gamma = 1/3), then the point sits in the middle of the triangle.
@ danielhsu021202 By "pulling," do you mean the distance between the point and the vertices? Basically the shorter the distance, the stronger the "pulling"? That is a really interesting way to think of it. I really appreciate your sharing.
I’m wondering how does it work intuitively if outside the triangle
@Songbird94 I think if a point is outside the triangle, then at least one of alpha, beta, or gamma is going to be negative. And intuitively, it's like the contribution of vertices A, B, and/or C is "pulling" the point outside the triangle. There's probably a bunch of other tests that the programs we write will use to evaluate if a point is inside or outside a triangle
Barycentric coordinates are especially useful for showing triangle geometry properties that do not depend on the angles - including Ceva's theorem, Menelaus's theorem, Routh's theorem, and Stewart’s theorem. Another cool application is in solving math olympiad geometry problems - this paper explains some of the theory and example problems: https://web.evanchen.cc/handouts/bary/bary-full.pdf
It's kind of interesting how you could use this to prove that 3 points define a plane; we can specify any point on the plane as a linear combination of the three points of this triangle (with the sum of the coefficients always being fixed at 1). Even though from a linear algebra perspective, the plane is 2-dimensional, it still takes 3 points to define the plane which means we have one dimension of freedom, which in this case is the sum of the coefficients. We could also fix the sum of the coefficients as 0, or -1, or pi, or whatever other number we want and still get a coordinate system, albeit one that's less relevant to the original triangle we were working with. One semi-interesting thing is that if we fix the sum as c, we can recover the barycentric coordinates of a copy of the triangle scaled by c by multiplying each point by c and restoring the sum of the coefficients to be 1.
Barycentric coordinates have an interesting connection to convexity, since all points in the triangle are a convex combination of the vertices (i.e. all alpha, beta and gamma must be positive for the point to lie in the triangle). Since their sum is bounded by one, if all are positive, then they will all be between 0 and 1, which satisfies the definition of a convex combination. If any of these weights is negative, the point will lie outside of the triangle.
Super interesting, reminds me of how I used the shoelace theorem a lot to complete geometry math problems
Barycentric Coordinates feel like a very interesting and useful way of defining points on a triangle and seem to have many useful properties. I'm interested in knowing what other use cases they have as a quick search seemed to reveal that they were mostly used only within computer graphics.