It looked like in Illustrator there was control of the endpoints and tangent, but also a "magnitude" of the tangent that could be adjusted by making the tangent line longer. This would suggest there's another degree of freedom we didn't account for before. My guess is that we gained this extra parameter when we moved from 1D to 2D. This really cool page https://pomax.github.io/bezierinfo/#control describes 2D Bezier curves as separate cubic equations in x and y. This gives us 8 degrees of freedom, which we could map to x, y, tangent, and "magnitude" of the tangent at the start and end point.
adampahlavan
How is this C1 continuous? Isn't f2 discontinuous in the first derivative?
tulum
@adampahlavan We may need to treat $f_i,\ i=1, \cdots, n$ as separate points. So the lecture slide actually said $f_2$ is not continuous but $f_1$ is.
It looked like in Illustrator there was control of the endpoints and tangent, but also a "magnitude" of the tangent that could be adjusted by making the tangent line longer. This would suggest there's another degree of freedom we didn't account for before. My guess is that we gained this extra parameter when we moved from 1D to 2D. This really cool page https://pomax.github.io/bezierinfo/#control describes 2D Bezier curves as separate cubic equations in x and y. This gives us 8 degrees of freedom, which we could map to x, y, tangent, and "magnitude" of the tangent at the start and end point.
How is this C1 continuous? Isn't f2 discontinuous in the first derivative?
@adampahlavan We may need to treat $f_i,\ i=1, \cdots, n$ as separate points. So the lecture slide actually said $f_2$ is not continuous but $f_1$ is.