3D Vectors

TPemath, Aug 12, 2024

Table of Contents

  1. Quadric Surfaces
    1. Ellipsoids
    2. Paraboloids
    3. Hyperboloids

1.

Quadric Surfaces

The two objects we studied in 3-dimensional space so far are the [b]line[/b] and the [b]plane[/b]. [list] [*] The line has symmetric equations [math] \frac{x-x_0}{A} = \frac{y-y_0}{B} = \frac{z-z_0}{C} [/math]. [*] The plane has point-normal form [math] A(x-x_0) + B(y-y_0) + C(z-z_0) = 0 [/math] [/list] What other objects belong to 3-dimensional space?

  • 1. Ellipsoids
  • 2. Paraboloids
  • 3. Hyperboloids

1.1.

Ellipsoids

Definition
Given positive numbers [math]a,b,c[/math] an [b]ellipsoid[/b] is given by the equation[br][center][br][math]\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1.[/math][/center]
Parameters
Change [math]a[/math] from [math]1[/math] to [math]3[/math]. What happens to the object?
Change [math]b[/math] from [math]1[/math] to [math]2[/math]. What happens to the object?
Change [math]b[/math] from [math]2[/math] to [math]1.5[/math]. What happens to the object?
Change [math]c[/math] from [math]1[/math] to [math]3[/math]. What happens to the object?
– TPemath

1.2.

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1.3.

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Information