Mirrored curves

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We study how to bounce vectors off of curves.

Work in groups of 3–4, writing your answers on a separate sheet of paper.

Normal vectors

Consider the following line:

Find a vector normal to this line. Explain your reasoning.

Now consider the line $ax+by = c$ in $\R ^2$ . Find a vector normal to this line. Explain your reasoning.

Consider the equation: $\vec {n}\dotp (\vec {x}-\vec {p}) = 0$ Explain how this connects to finding normal vectors to lines in $\R ^2$ of the form: $ax + by = c$ In particular, you should explain what $\vec {n}$ , $\vec {x}$ , and $\vec {p}$ represent.

Quick! Tell me normal vectors for the following lines:

(a): $-3x+7y=11$
(b): $4y =8$
(c): $x=y$
(d): $y=-4x+1$

Reflecting off of lines

Now we will explore how mirrors reflects light.

Law of Reflection Light is reflected at the same angle as it arrived, as measured from a line perpendicular to the mirror. Draw a picture illustrating this fact.

Let’s see if we can explain why the Law of Reflection is true. We’ll address this in the next several problems.

Consider the following diagram:

Explain why the opposite angles $\alpha$ and $\beta$ must be equal.

Label more angles in your picture. Some pairs of angles will sum to $180^\circ$ . Use this to conclude that $\alpha = \beta$ .

Since a mirror simply reflects light, we see that the initial light beam makes the same angle with the line as the reflected light beam:

Adding a normal vector to the diagram above:

Explain why $\theta$ and $\varphi$ are equal.

Someone says that if $\vec {n}$ is a normal vector to a mirror, and $\vec {v}$ represents a light beam, then the reflected light beam is given by: $\vec {r}=\vec {v} - 2\proj _{\vec {n}}(\vec {v})$ Give two explanations verifying the formula above:

An explanation where you sketch $\vec {r}$ and show that it is a reasonable answer.
An explanation where you show (via a computation) that the angle which light is reflected is the same angle as it arrived, as measured from a line perpendicular to the mirror.

Now imagine the line below as a mirror and imagine vector $\vec {v}$ as a light-ray that strikes the mirror. Find the vector representing the reflection of $\vec {v}$ .

Reflecting off of curves

Reflecting off of mirrored surfaces doesn’t require calculus unless the mirror is curved.

Let $\vec {p}(t) = \vector {t,t^2}$ We claim that $\vec {n} = \vector {2t,-1}$ is a normal vector for this curve for any given $t$ . Confirm or deny this claim.

Suppose light rays $\vec {v}=\vector {0,-1}$ are hitting a mirrored surface described by: $\vec {p}(t) = \vector {t,t^2}$ Find a formula for the vector reflected off the surface for any given value of $t$ . Call this vector $\vec {r}(t)$ . Explain your reasoning.

For any given value of $t$ , $\vector {t,t^2}$ is a point on our curve. In the previous problem you found a vector $\vec {r}(t)$ describing a reflected light beams $\vec {v}=\vector {0,-1}$ . Consider the line: $\vecl (s) = \vector {t,t^2} + s \cdot \vec {r}(t)$

Find $a$ so that $\vecl (a)$ intersects the $y$ -axis.
What is the $y$ -value of $\vecl (a)$ ?
Do the questions directly above tell us? What does this have to do with telescopes, space-heaters, vanity mirrors, and eavesdropping devices?