Question

Name and sketch the graph of each of the following equations in three-space. $$ -x^{2}+y^{2}+z^{2}=0 $$

Short answer

The surface is a double cone with its axis along the x-axis.

Step-by-step solution

Identify the Surface Type

The given equation is \(-x^2 + y^2 + z^2 = 0\). Recognize that this is a quadratic equation in three-space, and it's a form of a quadratic surface. Specifically, notice the structure matches that of a cone, which can be rewritten and recognized by \(x^2 = y^2 + z^2\).

Recognize the Cone Orientation

The equation \(x^2 = y^2 + z^2\) indicates a cone opening along the x-axis. In contrast to standard cones, this structure implies that as x deviates from zero, either positively or negatively, y and z form a circular cross-section.

Sketch the Graph

To sketch the graph, it's essential to understand that for a fixed value of \(x\), the equation \(y^2 + z^2 = x^2\) represents a circle in the \(yz\)-plane. For \(x = 0\), there is a single point at the origin \((0, 0, 0)\). As \(|x|\) increases, the radius of the circle increases, generating a double cone centered around the origin.

Key concepts

Quadric Surfaces

Quadric surfaces are the locus of points that satisfy a second-degree polynomial equation in three variables: typically, these are combinations of the variables squared. Think of these surfaces as 3D counterparts of conic sections (like ellipses, parabolas, and hyperbolas) you might remember from 2D geometry. In 3D, they form new shapes such as spheres, paraboloids, hyperboloids, and more.

There are several kinds of quadric surfaces, based on the signs and coefficients of the terms in their equations:

• Ellipsoids
• Paraboloids
• Hyperboloids
• Cylinders
• Cones

Cones, like in the given exercise equation, are particularly interesting as they extend infinitely in both directions from their apex. This occurs because the cone is symmetric across its axis.

Conic Sections

Conic sections are the cross-sections of a cone sliced at different angles, resulting in different 2D figures. You may have encountered these before: circles, ellipses, parabolas, and hyperbolas. Each type of section corresponds to a specific slicing angle and configuration relative to the cone.

• Circle: A slice perpendicular to the cone's axis.
• Ellipse: An angled slice that does not pass through the apex.
• Parabola: A slice parallel to one side of the cone.
• Hyperbola: A steep slice through both halves of the cone.

In the context of our exercise, we are dealing with the three-dimensional form of these shapes. Specifically, you identified the surface as a cone, which inherently contains circles within its structure when sliced perpendicularly to its axis, as shown by the equation \(y^2 + z^2 = x^2\).

Conic sections serve as fundamental elements that help us analyze and visualize complex surfaces in mathematics and physics.

Equation Symmetry

Symmetry in equations is about understanding the balance and mirrored qualities of geometric shapes. In 3D graphing, this becomes crucial as it helps predict how a surface behaves across different planes and axes. Symmetry makes graphing neat and consistent, as it allows us to only calculate once and apply those findings across mirrored sections.

For the exercise equation \(-x^2 + y^2 + z^2 = 0\), symmetry is evident:

• The equation features symmetry about the x-axis since reversing x into negative keeps the form unchanged.
• It also shows symmetry in the y and z variables due to their interchangeable squared terms, meaning it behaves identically across the yz-plane.

Understanding such symmetries helps simplify the sketching process, as identical patterns can easily be mirrored across the axis without extra calculations.

Surface Sketching

Surface sketching in three dimensions is an essential skill to visualize and interpret mathematical surfaces. When sketching surfaces like cones, we use their defining characteristics identified in equations to outline critical features step-by-step. This ensures accuracy and helps in identifying the surface's shape and direction.

From the exercise solution, we learn to:

• Identify the axis along which the cone opens, here it's along the x-axis.
• Mark key points such as the vertex at the origin \((0,0,0)\).
• Consider cross-sections: For example, at different x-values, circles form as \(y^2 + z^2 = x^2\) depicts.

As you grow more comfortable with surface sketching, your ability to interpret 3D shapes and understand their real-life implications improves significantly. Don't rush the process; take your time to explore pathways, symmetries, and cross-sections for clear and sophisticated sketches.