Problem 53 - Calculus with Concepts in Calculus

Question

Sketch the quadric surface. \(4 x^{2}+y^{2}-z^{2}=16\)

Step-by-Step Solution

Answer

The surface is a hyperboloid of one sheet, centered at the origin and elongated along the y-axis.

Step 1: Identify the Form of Equation

The given equation is \(4x^2 + y^2 - z^2 = 16\). This matches the standard form of a hyperboloid of one sheet \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\). Recognize that a hyperboloid of one sheet can be rewritten by dividing the equation by a constant to isolate 1 on the right-hand side.

Step 2: Rewrite Equation in Standard Form

Divide both sides of \(4x^2 + y^2 - z^2 = 16\) by 16 to transform it into the standard form. This gives: \[\frac{x^2}{4} + \frac{y^2}{16} - \frac{z^2}{16} = 1\]. Here, \(a^2 = 4\), \(b^2 = 16\), and \(c^2 = 16\).

Step 3: Interpret Standard Form Parameters

From \(\frac{x^2}{4} + \frac{y^2}{16} - \frac{z^2}{16} = 1\), identify that \(a = 2\), \(b = 4\), and \(c = 4\). These parameters tell us the shape is elongated along the y-axis and has its centers of symmetry at the origin.

Step 4: Sketch the Surface

Start by sketching the typical shape of a hyperboloid of one sheet. A hyperboloid of one sheet resembles an hourglass figure. Draw the x, y, and z axes and show that the hyperbola opens in the direction of the z-axis with vertices at \((0, 0, \pm4)\) because \(c = 4\).

Step 5: Mark Axes Intersections

For each axis, determine the intersection points by setting the other variable to zero. For the x-axis, set \(y = 0, z = 0\) which gives intersection at \((\pm 2, 0, 0)\). For the y-axis, set \(x = 0, z = 0\) resulting in intersections at \((0, \pm 4, 0)\). For the z-axis, set \(x = 0, y = 0\) for intersections at non-real solutions due to \(z^2 < 0\) if \(x = 0\) and \(y = 0\), emphasizing the center symmetry.

Key Concepts

Hyperboloid of One Sheet

A hyperboloid of one sheet is a fascinating type of quadric surface in three dimensions. It is characterized by its distinctive hourglass shape. In mathematical terms, a hyperboloid of one sheet can be represented by the equation: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \]. Here, the plus signs indicate hyperbolic sections in the plane, while the minus sign corresponds to the axial alignment. The symmetry of the hyperboloid comes from its equal distortion around its central axis. The hyperboloid of one sheet can be understood by its parameters \(a\), \(b\), and \(c\), representing the intercepts along the x, y, and z axes respectively. A key physical property is that it opens along the axis corresponding to the variable with the negative sign, typically the z-axis.

• Vertices along the axis are located at the points \((0, 0, \pm c)\), offering important insight into its shape.
• Such surfaces are commonly found in structural forms like cooling towers and modern architecture, owing to their stability and aesthetic appeal.

3D Surface Sketching

Sketching a 3D surface like a hyperboloid of one sheet helps in visualizing its structure. Begin by drawing the x, y, and z axes, which serve as your spatial reference frame. The most recognizable feature of a hyperboloid of one sheet is its two central openings commonly referred to as 'necks'. These open symmetrically along the z-axis when examining the standard form of its equation.

• Start by marking the vertices at \((0, 0, \pm c)\).
• Draw curves that capture the opening of the surface cross-sections, using the symmetry along the depicted axes as guides.

This hyperboloid will look like an hourglass, widening outward from the center then constricting again toward the other end. Repeated sketching reinforces spatial understanding.

Conic Sections

The concept of conic sections is integral when working with hyperboloids like the one depicted in the equation \(4x^2 + y^2 - z^2 = 16\). A conic section is a curve that can be obtained by intersecting a plane with a double-cone. When intersected with specific orientations, various conic shapes are formed:

• Ellipses
• Circles (special case of ellipses)
• Parabolas
• Hyperbolas

In our particular hyperboloid of one sheet, the cross-sections parallel to the xy-plane will form ellipses, reflecting a symmetric and rounded shape in each slice.Each conic section is derived from a particular slicing angle and distance from the origin, which informs the overall topology of the surface itself.

Surface Intersections

Surface intersections play a major role in understanding how different objects interact and align with quadric surfaces. In the context of the hyperboloid of one sheet, intersections occur at strategic points along its axes. To ascertain these points, set each variable to zero individually, allowing us to know where the hyperboloid touches each axis:

• For the x-axis, set \(y = 0\) and \(z = 0\), which results in intersections at \((\pm a, 0, 0)\).
• Similarly, for the y-axis, when \(x = 0\) and \(z = 0\), intersections occur at \((0, \pm b, 0)\).

It is worth noting that for the z-axis, with \(x = 0\) and \(y = 0\), intersection does not occur due to the form of the equation, emphasizing that the surface fully encompasses the axis within its 'necks'.Understanding these intersections not only boosts spatial insight but holds practical value in real-world applications such as computer graphics and architectural design.