# Distance between parametric function and a point

Given a parametric function \begin{align*} x &= x_o + v_x t + \frac{1}{2} a_x t^2\\ y &= y_o + v_y t + \frac{1}{2} a_y t^2\end{align*}

and point $P$ at $(c, d)$ , find all point(s) on the function that are distance $R$ from point $P$, assuming that $R > 0$ and none of the constants equal $0$.

This is part of a programming project (that I'm doing on my own, not for homework), so the answer needs to be obtainable using only expressions. Graphically solving the problem won't be possible.

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Welcome to math.SE. If this is a homework, then please add (homework) tag. – user2468 Mar 29 '12 at 19:24
Not homework. Personal project, for fun. – Jon W Mar 30 '12 at 18:52

I'm note sure if (homework).

Hint:

I'm assuming that we're given $R, c, d, x_0, v_x, a_x, y_0, v_y, a_y.$

The squared distance between $P = (c,d)$ and $(x,y)$ is given by $$R^2 = (x - c)^2 + (y - d)^2 \tag{1}$$ Substitute the definition of $$x = x_o + v_x * t + \frac{1}{2} * a_x * t^2 \tag{2} \\ y = y_o + v_y * t + \frac{1}{2} * a_y * t^2$$ in $(1)$

You're left with a polynomial in $t.$ Solve for $t,$ this will give you different values of $t = \{t_1, \ldots, t_4\}.$ Substitute each $t_i$ in $(2)$ to get different $(x,y)$ points.

Update # 1:

I'm too lazy to LaTeX the following equation:

        2          2   4                            3
(0.25 ay  + 0.25 ax ) t  + (1.0 vy ay + 1.0 vx ax) t

2                                       2               2
+ (vx  + 1.0 xo ax - 1.0 ax c - 1.0 ay c + vy  + 1.0 yo ay) t

2       2       2
+ (-2. vy c + 2. yo vy + 2. xo vx - 2. vx c) t + xo  + 2. c  - 1. R

2
- 2. xo c - 2. yo   = 0


Anyways, that's the degree $4$ polynomial in $t.$ To find a closed-form expressions for the roots $$\{t_1 = \cdots, t_2 = \cdots, t_3 = \cdots, t_4 = \cdots\},$$ you will need to do a lot of algebra on this polynomial. For example this.

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The problem is that because this is for a program, each possible value of $t$ must be defined as a single expression ($t1 = ..., t2 = ...$) using $R,c,d,x_0,v_x,a_x,y_0,v_y,a_y$. The numerical values for these constants are unknown, since the values will vary when this is used in the program. The problem that I'm facing is that when I take the expanded polynomial and solve for $t$ using my graphing calculator, it is unable to find a solution. – Jon W Mar 30 '12 at 18:46
Thanks. I found this thing, which I think I might be able to use. Since the massive equation basically follows $A * t^4 + B * t^3 + C * t^2 + D * t + E = 0$, I could probably use the method described to solve the problem. All of the constants ARE numerical values, so I can just plug them in to find the numerical values for A, B, C, D, and E. – Jon W Mar 30 '12 at 19:09
@JonW Indeed. The big expression above gives you $A, B, C, D, E.$ You're on the right path. – user2468 Mar 30 '12 at 19:11