Parabolic reflection

Parametric equations

The parabola $y^2=4px$ opens to the right. The focus is $F=(p,0)$. Rays leave $F$, meet the curve, and continue horizontally to the right, parallel to the axis.

• $p$ is the focal length, the distance from the vertex to the focus $F=(p,0)$
• At a point $P(x,y)$ on the curve, the reflected ray has direction $(1,0)$, to the right along the $x$-axis

Interaction

• Focal length p: change how wide the parabola opens; the focus moves with $p$
• Ray count: add or remove the rays that leave the focus
• Scan speed ω: change how fast the hit points stretch up and down together; the reflected rays stay horizontal

What to notice

• Rays that leave the focus are parallel to each other after they reflect.
• Changing $p$ moves the opening and the focus. The reflected rays stay parallel; they do not gather to a point.
• During the scan, every reflected ray stays parallel to the axis while the hit heights stretch and shrink together.

Related

• Conic envelope
• Focus locus
• Conics by eccentricity

Further reading

• Parabola (Wikipedia)