Problem 39 - Calculus with Concepts in Calculus

Question

Sketch the quadric surface. \(x^{2}+2 y^{2}+3 z^{2}=6\)

Step-by-Step Solution

Answer

The quadric surface is an ellipsoid.

Step 1: Identify the Type of Quadric Surface

The general equation of a quadric surface is given by \[Ax^{2} + By^{2} + Cz^{2} + Dz + Ey + Fx = G \]In this exercise, the given equation is \[x^2 + 2y^2 + 3z^2 = 6\]This corresponds to \[A = 1, B = 2, C = 3, D = 0, E = 0, F = 0, G = 6\]Recognizing it matches the standard form of an ellipsoid, \[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\], where the coefficients are all positive, confirms this is the equation of an ellipsoid.

Step 2: Standardize the Equation

Divide the whole equation by 6 to convert it into the standard form, \[\frac{x^2}{6} + \frac{y^2}{3} + \frac{z^2}{2} = 1\]. This puts the equation into the form analogous to the ellipsoid equation, where \(a^2 = 6\), \(b^2 = 3\), and \(c^2 = 2\).

Step 3: Interpret the Semiaxes Lengths

The semiaxes lengths are the square roots of the denominators: \(a = \sqrt{6}\), \(b = \sqrt{3}\), \(c = \sqrt{2}\). These indicate the ellipsoid stretches farther in the direction of the x-axis (radius \(\sqrt{6}\)), followed by the y-axis (radius \(\sqrt{3}\)), and the z-axis (radius \(\sqrt{2}\)).

Step 4: Sketch the Ellipsoid

Draw the principal axes in the coordinate space. Along the x-axis, mark the points \(+\sqrt{6}\) and \(-\sqrt{6}\), along the y-axis, mark \(+\sqrt{3}\) and \(-\sqrt{3}\), and along the z-axis, mark \(+\sqrt{2}\) and \(-\sqrt{2}\). Draw a smooth, rounded shape connecting these points symmetrically, representing the ellipsoid.

Key Concepts

Ellipsoids

An ellipsoid is a three-dimensional geometric surface that extends the concept of an ellipse into three dimensions. Imagine stretching or compressing a sphere in various directions, and you get an ellipsoid.
A standard equation of an ellipsoid is: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\]Here, \(a\), \(b\), and \(c\) are the lengths of the semi-principal axes along the x, y, and z-axis, respectively. The ellipsoid's center is usually at the origin,
but it can be shifted in space by translating the axes of the equation.

• If all three axes are equal (\(a = b = c\)), the ellipsoid is a sphere.
• When two axes are equal and one is different, the shape resembles a squashed or stretched sphere known as a spheroid.
• If all three semi-principal axes are different, the result is a tri-axial ellipsoid.

Understanding the relative lengths of these axes gives insight into how the ellipsoid is shaped and oriented in space, making it a fundamental object in various fields of science and engineering.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double napped cone. The family of conic sections includes ellipses, parabolas, and hyperbolas, each reflecting a different type of intersection.
Ellipses, like the cross-section of an ellipsoid, arise from slicing through a cone at an angle. This leads to a symmetrical, oval-shaped curve that is closed and bounded.

• An ellipse results when the intersecting plane cuts through both nappes of the cone, but not at a steep enough angle to form a hyperbola.
• In the special case where the slicing plane is horizontal to the base, the ellipse becomes a perfect circle.

Each conic section has unique properties and equations. Ellipses can be used to model planetary orbits, while parabolas and hyperbolas have applications in physics and architecture.
Understanding these forms is crucial for comprehending more complex geometric and physical phenomena.

Analytic Geometry

Analytic geometry, also known as coordinate geometry, is the study of geometry using a coordinate system. It allows us to define geometric shapes algebraically and use these equations to solve geometric problems.
By using coordinates, we can express curves and surfaces like ellipsoids as sets of equations, making it easier to visualize and manipulate these shapes.

• In two dimensions, you often work with lines, circles, and parabolas, using coordinates (x, y).
• In three dimensions, the equations become more complex since they involve planes and surfaces like spheres and ellipsoids.

Analytic geometry bridges the algebraic and geometric perspectives.
It provides a powerful tool to solve real-world problems involving distances, intersections, and other spatial relationships. This method is foundational for calculus and has applications ranging from computer graphics to physics.