Normal Lines on a Surface Examples 1

Recall from the Normal Lines on a Surface page that if we have a two variable real-valued function $z = f(x, y)$ and $P(x_0, y_0, z_0)$ is a point on the surface generated by this function, then the normal line at $P$ is the line that passes through $P$ and is perpendicular to the tangent plane at $P$. This normal line is given parametrically by:

(1)

\begin{align} \left\{\begin{matrix} x = x_0 + \frac{\partial}{\partial x} f(x_0, y_0) t\\ y = y_0 + \frac{\partial}{\partial y} f(x_0, y_0) t\\ z = z_0 - t \end{matrix}\right. \quad -\infty < t < \infty \end{align}

We will now look at some examples of computing the normal line to a surface at a specific point.

Example 1

Find parametric equations for the normal line to the paraboloid $f(x, y) = 4 - x^2 - y^2$ at the point $(1, 1, 2)$.

We first calculate the partial derivatives of $f$. We have that:

(2)

\begin{align} \quad \frac{\partial f}{\partial x} = -2x \quad , \quad \frac{\partial f}{\partial y} = -2y \end{align}

Therefore, $\frac{\partial}{\partial x} f(1, 1) = -2$ and $\frac{\partial}{\partial y} f(1, 1) = -2$. Therefore the normal line to the paraboloid at $(1, 1, 2)$ is given parametrically by:

(3)

\begin{align} \left\{\begin{matrix} x = 1 - 2t\\ y = 1 -2t\\ z = 2 - t \end{matrix}\right. \quad -\infty < t < \infty \end{align}

Example 2

Find parametric equations for the normal line to the circular cylinder $x^2 + y^2 = 1$ at $(1, 0, 2)$ (Hint: You do not need to calculate partial derivatives in order to find the parametric equations for this line).

Notice that the circular cylinder $x^2 + y^2 = 1$ is formed from vertical rulings that are parallel to the $z$-axis. The vector $(0, 0, 1)$ is parallel to the $z$-axis, and so any vector of the form $(a, b, 0)$ is perpendicular to the $z$ axis since by the dot product we have that:

(4)

\begin{align} \quad (a, b, 0) \cdot (0, 0, 1) = 0 \end{align}

We want to choose a vector that is also in the same direction as the point $(1, 0, 5)$, so choose the vector $(1, 0, 0)$. Therefore we have that the parametric equations of the normal line at $(1, 0, 2)$ is given by:

(5)

\begin{align} \left\{\begin{matrix} x = 1 + t\\ y = 0\\ z = 2 \end{matrix}\right. \quad -\infty < t < \infty \end{align}