It makes sense intuitively because as the point gets close to x_i, both function's values get larger, though I don't know how to mathematically prove it...
ngandhi
Yeah this makes sense, because mathematically, the area of a triangle is 1/2 * base * height. Both triangles in that equation have the same base and the 1/2 factor, so you can cancel it and all you get left with is just the ratio of the heights, which is what was done in the previous slide.
keenan
@nghandi Nice explanation. :-)
CMUScottie
yes, it is just the same as the ratio of height, because the area is proportional to height x base, and they share the same base.
FeiFeiFei
Yes. I buy it. The area of triangle is linear to its height and also the two triangles share the base, hence the ratios of height and area are basically the same.
tib
Yes, as mentioned above, since the base values are fixed in each change, the changing variable that varies the area is the height, which is proportional to the previous definition of the interpolation.
emmurphy
Yes, the equation is indeedly similar to the previous explanation. Because the area of triangle is 1/2 * height * base
Isaaz
Yes, because we can cancle out the base and constant to simplify the function to be the same as the one in last method.
graphicstar11
yes this makes sense since these triangles all share the same base so the the area is changing proportional to height with the triangle area being 1/2(hb)
atarng
If you write out the area formula you'll find that the 1/2*b parts cancel out, leaving you with just a ratio of heights, which is what we did on the last slide. Doesn't seem very practical to do this instead of the heights ratio since you're just undoing your own work.
keenan
@atarng Actually in terms of writing simple code, this is a pretty slick way to express the barycentric coordinates. YMMV.
It makes sense intuitively because as the point gets close to x_i, both function's values get larger, though I don't know how to mathematically prove it...
Yeah this makes sense, because mathematically, the area of a triangle is 1/2 * base * height. Both triangles in that equation have the same base and the 1/2 factor, so you can cancel it and all you get left with is just the ratio of the heights, which is what was done in the previous slide.
@nghandi Nice explanation. :-)
yes, it is just the same as the ratio of height, because the area is proportional to height x base, and they share the same base.
Yes. I buy it. The area of triangle is linear to its height and also the two triangles share the base, hence the ratios of height and area are basically the same.
Yes, as mentioned above, since the base values are fixed in each change, the changing variable that varies the area is the height, which is proportional to the previous definition of the interpolation.
Yes, the equation is indeedly similar to the previous explanation. Because the area of triangle is 1/2 * height * base
Yes, because we can cancle out the base and constant to simplify the function to be the same as the one in last method.
yes this makes sense since these triangles all share the same base so the the area is changing proportional to height with the triangle area being 1/2(hb)
If you write out the area formula you'll find that the 1/2*b parts cancel out, leaving you with just a ratio of heights, which is what we did on the last slide. Doesn't seem very practical to do this instead of the heights ratio since you're just undoing your own work.
@atarng Actually in terms of writing simple code, this is a pretty slick way to express the barycentric coordinates. YMMV.