Question

Is it possible to solve for `u` and `v`, given the following matrix multiplication chain?

I am attempting to determine the intersections between a parametric Bezier surface, and line `L=P_0+tV`, where `P` is a point that lies on the line, and `V` is a vector describing the line's direction.

The following image's source presents a seemingly convenient matrix form. The equation in the image represents the calculation of a point that lies on the surface, given the parameters `u` and `v`.

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I figured the next logical step would be to substitute `P(u,v)` for the equation of the line given above, but I am unsure how to proceed.

I have tried to isolate the `u` and `v` coefficient matrices, but to no avail.

Is it possible to solve for `u` and `v`? Are there any alternatives?

Answer 1

See below.

Explanation:

After matrix multiplication we obtain

`P(u,v)=(x(u,u),y(u,v),z(u,v))`

The line is defined as

`L->P=P_0+t V`

The intersections are obtained solving

`{(x(u,v)=P_0^x+t V_x),(y(u,v)=P_0^y+t V_y),(z(u,v)=P_0^z+t V_z):}`

Three equations with three unknowns. The solutions can be obtained using an iterative procedure like Newton-Raphson.

`xi_(k+1} = xi_k-J(xi_k)^(-1)g(xi_k)`

where

`xi_k=(u_k,v_k,t_k)`
`g(xi_k)=((x(u_k,v_k)-P_0^x-t_k V_x),(y(u_k,v_k)-P_0^y-t_k V_y),(z(u_k,v_k)-P_0^z-t_k V_z))`

`J(xi_k)= (((partial x)/(partial u)(u_k,v_k),(partial x)/(partial v)(u_k,v_k),-V_x),((partial y)/(partial u)(u_k,v_k),(partial y)/(partial v)(u_k,v_k),-V_y),((partial z)/(partial u)(u_k,v_k),(partial z)/(partial v)(u_k,v_k),-V_z))`

Calling now

`P(u,v)=U^TcdotC^TcdotPicdotCcdotV`

where

`U=(1,u,u^2,u^3)^T`
`V=(1,v,v^2,v^3)^T`

we have

`(partial P)/(partial u) = ((dU)/(du))^TcdotC^TcdotPicdotCcdotV`
`(partial P)/(partial v) = U^TcdotC^TcdotPicdotCcdot(dV)/(dv)`