How to calculate a straight with a position vector (x,y) and a direction vector (x,y)

Question

I want calculate the reflection from a straight R on the surface from the straight BC:

The triangle with the position vectors A,B,C are given. As well the position vector P and the direction vector R.

what I can calculate:

Now I can calculate the intersection point D(x,y) and the reflection vector (as a direction vector R') and I know the x coordinate from the point P'(x,?) because the x value from the point P and P' are equals.

what I cannot calculate:

I need from the point P' the y coordinate. Because I need position vectors for drawing the reflected line with python matplotlib.

How can I get the y coordinate from the point P' with the given parameters?

The reflected vector will go from D(x,y) to P'(x,y)

matplotlib reflection

Thanks you!

Answer 1

It seems that you have already solved the problem of reflection, so we can ignore everything except the point $\mathbf D$ and vector $\mathbf R',$ which you have computed, and the desired result $\mathbf P' = (x_P, y_{P'}),$ where $x_P$ is the $x$-coordinate of $P$ and is already given.

Take the coordinates $\mathbf D = (x_D,y_D)$ and $\mathbf R' = (r'_x, r'_y).$ The line in the direction $\mathbf R'$ from $\mathbf D$ then can be described by the equations \begin{align} x &= x_D + tr'_x,\\ y &= y_D + tr'_y,\\ \end{align} where $t$ is a variable real number.

You can then find the value of $t$ that will given you the point $\mathbf P'$ by solving this equation: $$ x_P = x_D + tr'_x. $$ Everything here is known except $t,$ so this is a simple linear equation to solve: $$ t = \frac{x_P - x_D}{r'_x}.$$

Take this value and plug it into the formula for $y$: $$ y_{P'} = y_D + tr'_y. $$

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For some ways of setting up this problem, you could get a negative value of $t,$ for example if $\mathbf P$ was much further to the left and the vector $\mathbf R$ pointed downward to the right instead of to the left to reach $\mathbf D.$ In that case there would not be a valid solution, and a good diagram would confirm that there cannot be one. But in the figure as you have drawn it, $t$ will be positive.

Answer 2

Work in polar coordinates with $D$ as the origin. You can compute the angle that BC lies at. If you add or subtract (depending on which end of $BC$ you choose) $\frac \pi 2$ you get the angle of the normal, which leaves $D$ in an upward right direction perpendicular to $BC$. Call a point on it $N$. Then use angle of incidence equals angle of reflection. You have $\angle PDN=\angle NDP'$