It looks like all smooth curve interpolation schemes have to be written as matrices. In other words, they have basis functions we can use to easily compute any P(t). I wonder if this a requirement, and if so, is it because the graphics pipeline has to be super fast? That is my reasoning.
sheaconlon
@orkun1675 I think any interpolation scheme is going to implicitly define a space of functions. If the space has n dimensions (in this case 4), then it should be possible to identify n basis functions to span the space. Since we are supplying the values of the terms of the equation (the powers of t) as inputs to the matrix multiplication, the resulting system is then linear. So I think this is just a natural consequence and it is linear because we purposely make it linear.
It looks like all smooth curve interpolation schemes have to be written as matrices. In other words, they have basis functions we can use to easily compute any P(t). I wonder if this a requirement, and if so, is it because the graphics pipeline has to be super fast? That is my reasoning.
@orkun1675 I think any interpolation scheme is going to implicitly define a space of functions. If the space has n dimensions (in this case 4), then it should be possible to identify n basis functions to span the space. Since we are supplying the values of the terms of the equation (the powers of t) as inputs to the matrix multiplication, the resulting system is then linear. So I think this is just a natural consequence and it is linear because we purposely make it linear.