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How to find the intersection of a parametric line and a sphere

Discussion in 'Mathematics' started by jtd121200, Oct 8, 2018.

  1. jtd121200

    jtd121200 Guest

    For a homework assignment, one of the questions requires finding the points where a parametric line in vector form intersects with a sphere whose radius changes.

    The line $r(s)$ is denoted by $\textbf{OP} + s\textbf{v}$, where $P = (x,y,z)$, and $\textbf{v} = \langle a,b,c\rangle$ (In the problem, there are actual values for the variables, but I'd like to look at it as a more general solution to help actually understand the problem).

    The sphere has a radius $t$, which grows as $t$ increases.

    The part of the problem that I'm having trouble understanding is how to find the points of intersection between the sphere and $r(s)$. I know that all intersections will fall on the line, but I'd like to find which value of $t$ will result in the first intersection, when $r(s)$ is tangent to the sphere's surface. When I visualize a similar problem in $\mathbb{R}^2$, it becomes easier to see when a circle intersects a line, but I have trouble finding out how to apply this in $\mathbb{R}^3$, and with a line that is parametric.

    Any nudges in the right direction would greatly help.

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