Question

Consider the following equations of quadric surfaces. a. Find the intercepts with the three coordinate axes, when they exist. b. Find the equations of the \(x y-, x z^{-}\), and \(y z\) -traces, when they exist. c. Sketch a graph of the surface. $$2 y-\frac{x^{2}}{8}-\frac{z^{2}}{18}=0$$

Short answer

Based on the given surface equation, $$2y - \frac{x^2}{8} - \frac{z^2}{18} = 0$$, sketch and analyze the surface graph, considering its traces and intercepts.

Step-by-step solution

Finding the intercepts

To find the intercepts, we will set the other two variables to zero and solve the equation. For \(x\)-axis intercept, let \(y = z = 0\): $$-\frac{x^2}{8} = 0 \Rightarrow x^2 = 0 \Rightarrow x = 0$$ There are no \(x\)-axis intercepts. For \(y\)-axis intercept, let \(x = z = 0\): $$2y = 0 \Rightarrow y = 0$$ There are no \(y\)-axis intercepts. For \(z\)-axis intercept, let \(x = y = 0\): $$-\frac{z^2}{18} = 0 \Rightarrow z^2 = 0 \Rightarrow z = 0$$ There are no \(z\)-axis intercepts.

Finding the traces

Now, we will find the equations for the XY-, XZ-, and YZ- traces. For the XY-trace, let \(z = 0\): $$2y - \frac{x^2}{8} = 0$$ For the XZ-trace, let \(y = 0\): $$-\frac{x^2}{8} - \frac{z^2}{18} = 0$$ For the YZ-trace, let \(x = 0\): $$2y - \frac{z^2}{18} = 0$$ So, the traces are: XY: \(2y - \frac{x^2}{8} = 0\) XZ: \(-\frac{x^2}{8} - \frac{z^2}{18} = 0\) YZ: \(2y - \frac{z^2}{18} = 0\)

Sketch a graph of the surface

To sketch the graph of the surface, we can analyze the given equation and its traces: Surface equation: $$2y - \frac{x^2}{8} - \frac{z^2}{18} = 0$$ This equation represents a hyperbolic paraboloid opening upward along the \(y\)-axis. Plotting the XY-, XZ-, and YZ- trace equations, we can sketch the surface by joining the traces and considering the direction of the surface (upward along the \(y\)-axis). Overall, the surface is a hyperbolic paraboloid with no intercepts on any of the axes.

Key concepts

Intercepts

Intercepts are the points where a surface intersects with the coordinate axes, and they are found by setting two variables to zero and solving for the third. In the exercise given, when checking for intercepts along each axis for the equation \(2y - \frac{x^2}{8} - \frac{z^2}{18} = 0\),we set two of the variables to zero and solved for the third. However, in all cases:

• For the x-axis: setting \(y = 0\) and \(z = 0\) leads to \(x = 0\).
• For the y-axis: setting \(x = 0\) and \(z = 0\) leads to \(y = 0\).
• For the z-axis: setting \(x = 0\) and \(y = 0\) leads to \(z = 0\).

In each scenario, the intercept is zero, indicating that the surface does not intersect the axes at any point other than the origin. This happens because the equation does not allow any value for which only one variable is non-zero.

Traces

Traces are created by slicing the surface with planes parallel to the coordinate planes (XY-, XZ-, and YZ-traces). For each trace, we set the coordinate not in the plane to zero and solve the resulting equation. In our exercise, finding each trace was straightforward, as follows:

• **XY-trace**: Setting \(z = 0\), we have the equation \(2y - \frac{x^2}{8} = 0\). This represents a parabola with the y-axis as its axis of symmetry.
• **XZ-trace**: Setting \(y = 0\), the equation becomes \(-\frac{x^2}{8} - \frac{z^2}{18} = 0\), which simplifies to a hyperbola.
• **YZ-trace**: Setting \(x = 0\), we get \(2y - \frac{z^2}{18} = 0\), another parabola but now with the y-axis as its axis of orientation.

Each trace provides a glimpse into the behavior of the surface, which helps in visualizing the 3D shape.

Hyperbolic Paraboloid

A hyperbolic paraboloid is a type of quadric surface that features saddle points, resembling a Pringles potato chip or a "saddle" shape. From the equation \(2y - \frac{x^2}{8} - \frac{z^2}{18} = 0\), we can identify this surface as a hyperbolic paraboloid because:

• The equation contains both positive and negative quadratic terms.
• One variable (\(y\) here) is linear, indicating the axis along which the surface opens, heading upwards.

The traces we found also support this classification:

• The parabolas in the XY- and YZ-traces each show the upward opening in a specific plane.
• The hyperbola found in the XZ-trace reflects the saddle point nature of the surface, exhibiting curves in opposite directions.

Overall, understanding the hyperbolic paraboloid through traces and intercepts allows us to visualize its intricate 3D structure.