I am looking at a book about ray-tracing and there is this equation about a ray intersecting a sphere.

It shows this equation:

dot((A + t*B - C),(A + t*B -C)) = R*R

Which I think I understand. Now by moving terms of the equation to the left hand side it becomes this:

t*t*dot(B,B + 2*t*dot(B,A-C) + dot(A-C,A-C) - R*R=0

Could somebody explain what steps are taken to achieve this? I understand moving - R*R to the left and then getting 0 after = but that's about it.

I think moving the terms is necessary to get the t out of the dot product as t is the one variable that is unknown for this case.

Hope this falls under Arithmetic.

Help much appreciated.

It shows this equation:

dot((A + t*B - C),(A + t*B -C)) = R*R

Which I think I understand. Now by moving terms of the equation to the left hand side it becomes this:

t*t*dot(B,B + 2*t*dot(B,A-C) + dot(A-C,A-C) - R*R=0

Could somebody explain what steps are taken to achieve this? I understand moving - R*R to the left and then getting 0 after = but that's about it.

I think moving the terms is necessary to get the t out of the dot product as t is the one variable that is unknown for this case.

Hope this falls under Arithmetic.

Help much appreciated.

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