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Magazine Chapter 12: Problem 20
Use traces to sketch and identify the surface. $$ x=y^{2}-z^{2} $$
Short Answer
Expert verified
The surface is a hyperbolic paraboloid.
Step by step solution
01
Understand the Equation
The given equation is \(x = y^2 - z^2\). This is a quadratic equation involving two variables, \(y\) and \(z\), with \(x\) as the result. Such equations typically describe a type of surface known as a quadric surface.
02
Recognize the Surface
The equation \(x = y^2 - z^2\) resembles the format of a hyperbolic paraboloid, which can be written in general form as \(x = A y^2 - B z^2\). Here, both \(A\) and \(B\) are equal to 1.
03
Trace in the xy-plane (z = 0)
Set \(z = 0\) and analyze the trace: \(x = y^2\). This is the equation of a parabola opening in the positive \(x\)-direction.
04
Trace in the xz-plane (y = 0)
Set \(y = 0\) and analyze the trace: \(x = -z^2\). This is the equation of a parabola opening in the negative \(x\)-direction.
05
Trace in the yz-plane (x = 0)
Set \(x = 0\) and analyze the trace: \(y^2 = z^2\). This is the equation representing two intersecting lines, \(y = z\) and \(y = -z\), which intersect at the origin.
06
Sketch the Surface
Combine the information from each of the traces: The traces indicate a surface with opposite parabolic openings (in \(x\)-direction for both \(y\)-axis and \(z\)-axis) and intersecting lines in the \(y = z\) and \(y = -z\) planes. This confirms the surface is a hyperbolic paraboloid, which looks like a saddle.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadric Surface
In the realm of three-dimensional geometry, a quadric surface is a second-degree equation in three variables, typically written in the form:
- Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fxz + Gx + Hy + Iz = J
Traces
To better understand the shape and orientation of a quadric surface, we use traces, which are essentially cross-sections of the surface through the coordinate planes. By setting each variable to zero in turn, you can gain insight into how the surface behaves:
- Trace in the xy-plane (\(z = 0\)): The equation simplifies to \(x = y^2\), forming a standard parabola opening along the positive \(x\)-axis.
- Trace in the xz-plane (\(y = 0\)): Here, we have \(x = -z^2\), another parabola, but this time it opens along the negative \(x\)-axis.
- Trace in the yz-plane (\(x = 0\)): The equation becomes \(y^2 = z^2\), resulting in two intersecting lines, specifically \(y = z\) and \(y = -z\), crossing at the origin.
Intersecting Lines
In the context of quadric surfaces, intersecting lines occur when we evaluate traces like the yz-plane: \(y^2 = z^2\). Solving for \(y\), we find that \(y = z\) and \(y = -z\). These lines don鈥檛 curve or meet at a single point unless \(y\) and \(z\) are both zero, where they intersect the x-axis at the origin.
This intersection feature is a crucial component in understanding how the surface behaves, showcasing lines that intersect, thus indicating the saddle nature of a hyperbolic paraboloid.
This intersection feature is a crucial component in understanding how the surface behaves, showcasing lines that intersect, thus indicating the saddle nature of a hyperbolic paraboloid.
Parabola
A parabola is a U-shaped curve defined by a quadratic equation such as \(y = x^2\). It is the simplest conic section that occurs in various branches of mathematics and has several characteristics:
- When we set \(z = 0\), our equation becomes \(x = y^2\), revealing a parabola opening rightward in the xy-plane.
- In contrast, setting \(y = 0\) gives \(x = -z^2\), showing a parabola opening leftward in the xz-plane.
