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Chapter 12: Problem 9

(a) Find and identify the traces of the quadric surface \(x^{2}+y^{2}-z^{2}=1\) and explain why the graph looks like the graph of the hyperboloid of one sheet in Table \(1 .\) (b) If we change the equation in part (a) to \(x^{2}-y^{2}+z^{2}=1\) how is the graph affected? (c) What if we change the equation in part (a) to \(x^{2}+y^{2}+2 y-z^{2}=0 ?\)

Short Answer

Expert verified
The graph is a hyperboloid of one sheet; changing x, y, or z flips orientation or shifts the graph.

Step by step solution

01

Trace in the xy-plane

To find the trace in the xy-plane, set \(z = 0\) in the equation. The equation becomes \(x^2 + y^2 = 1\). This represents a circle of radius 1 centered at the origin in the xy-plane.
02

Trace in the xz-plane

To find the trace in the xz-plane, set \(y = 0\) in the equation. The equation becomes \(x^2 - z^2 = 1\). This is a hyperbola opening along the x-axis in the xz-plane.
03

Trace in the yz-plane

To find the trace in the yz-plane, set \(x = 0\) in the equation. The equation becomes \(y^2 - z^2 = 1\), which is a hyperbola opening along the y-axis in the yz-plane.
04

Identify the Surface

The nature of the traces indicates that the surface is a hyperboloid of one sheet, characterized by its hyperbolic traces in one direction and circular trace in the plane perpendicular to that direction.
05

Varying the Equation (b)

When changing the equation to \(x^2 - y^2 + z^2 = 1\), the roles of the y-component and z-component switch. In this case, traces will be circular in the xz-plane and hyperbolic in the xy-plane and yz-plane, representing a hyperboloid of one sheet with a change in orientation.
06

Varying the Equation (c)

For the equation \(x^2 + y^2 + 2y - z^2 = 0\), first complete the square for the y-term: \((y+1)^2 - 1\). The equation becomes \(x^2 + (y+1)^2 - 1 - z^2 = 0\), thus \(x^2 + (y+1)^2 - z^2 = 1\). This still represents a hyperboloid of one sheet, but shifted in the y-direction by -1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Traces of Surfaces
In the realm of quadric surfaces, understanding traces is crucial for visualizing three-dimensional shapes. A trace is essentially a cross-section, obtained by slicing a surface with a plane. By examining traces, you can identify and differentiate among various quadric surfaces. For the surface represented by the equation \(x^2 + y^2 - z^2 = 1\), each trace relates to a specific plane.

For example:
  • Setting \(z = 0\), we discover the trace on the xy-plane is a circle, defined by \(x^2 + y^2 = 1\).
  • Setting \(y = 0\), the trace in the xz-plane forms a hyperbola, expressed by \(x^2 - z^2 = 1\).
  • Setting \(x = 0\), the trace in the yz-plane is also a hyperbola, given by \(y^2 - z^2 = 1\).
Traces unravel the geometry lurking in quadric surfaces, serving as a foundational tool for surface identification.
Hyperboloid of One Sheet
One of the intriguing quadric surfaces is the hyperboloid of one sheet. This structure boasts a rich geometric form featuring:
  • Hyperbolic traces in two planes, representing the curves that bow outwards.
  • A circular trace in the third plane, displaying symmetry around a central axis.
The equation \(x^2 + y^2 - z^2 = 1\) perfectly exemplifies this surface. The two hyperbolic traces manifest along the xz and yz planes, while the circular trace in the xy-plane showcases the unique combination of symmetry and curvature. By understanding these characteristics, one can easily recognize the hyperboloid of one sheet among other complex 3D shapes.
Equation Transformation
Transforming an equation provides insight into how surface orientations change. Altering the original equation to \(x^2 - y^2 + z^2 = 1\) flips the role of the y and z terms.
  • The traces become circular in the xz-plane (\(x^2 + z^2 = 1\)).
  • Hyperbolic traces occur in both the xy-plane and the yz-plane.
Such transformation may look simple but reveals how altering terms affects surface orientation and symmetry. It highlights the elegant mathematical interplay that governs the geometry of such surfaces.
Coordinate Planes
Coordinate planes鈥攑rimarily the xy-plane, xz-plane, and yz-plane鈥攁ssist in dissecting complex surfaces into more comprehensible parts.
By separately analyzing each slice a plane imposes on a surface, you gain clarity on the geometry present. Coordinate planes allow:
  • Decomposition of equations into simpler forms like circles or hyperbolas.
  • Identification of symmetry and key directional features.
For any quadric surface, comprehending how it intersects each coordinate plane provides critical insight into its overall structure, easing both visualization and calculation tasks.
Surface Identification
Surface identification entails understanding the equation and its associated traces to determine the type of quadric a surface represents. Often, specific configurations of traces align with standard forms like:
  • Circular traces indicating rotational symmetry.
  • Hyperbolic traces pointing to a hyperboloid structure.
By linking traces with known surfaces, like the hyperboloid of one sheet, and recognizing how transformations shift or rotate these shapes, you masterfully navigate the landscape of three-dimensional geometry. Each surface's uniqueness lies not only in its equation but also in how it's dissected, revealing secrets of its form.

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Most popular questions from this chapter

Prove that \((\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=\left|\begin{array}{ll}{\mathbf{a} \cdot \mathbf{c}} & {\mathbf{b} \cdot \mathbf{c}} \\ {\mathbf{a} \cdot \mathbf{d}} & {\mathbf{b} \cdot \mathbf{d}}\end{array}\right|\)

\(6-12\) Find parametric equations and symmetric equations for the line. The line through the origin and the point \((4,3,-1)\)

Find a vector equation and parametric equations for the line. The line through the point \((1,0,6)\) and perpendicular to the plane \(x+3 y+z=5\)

Find an equation of the plane. The plane through the point \((3,-2,8)\) and parallel to the plane \(z=x+y\)

(a) Find parametric equations for the line of intersection of the planes and (b) find the angle between the planes. $$ 3 x-2 y+z=1, \quad 2 x+y-3 z=3 $$
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