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Parameterizing an ellipse involves expressing its coordinates using a parameter, commonly denoted as \( t \). This parameter is akin to an angle in trigonometry and is instrumental in defining the position on the ellipse as \( t \) varies. For an ellipse centered at the origin with its axes aligned with the coordinate axes, the standard parameterization is:\[(x, z) = (a \cos(t), b \sin(t))\]Here, \( a \) and \( b \) represent the semi-major and semi-minor axes, respectively. This formula allows us to calculate the \( x \) and \( z \) coordinates for any position on the ellipse as the parameter \( t \) changes from 0 to \( 2\pi \). This parameterization takes advantage of trigonometric functions to describe the elliptical path, making it a versatile and straightforward way to describe ellipses in 2D or 3D space.

In the context of parameterizing curves like ellipses, adding a third dimension, such as the \( y \)-coordinate for a 3D parameterization, expands our understanding and application of the concept. A 3D coordinate for an ellipse may look like:\[(x(t), y(t), z(t)) = (a \cos(t), y_0, z_0 + b \sin(t))\]This indicates:

• \(x(t)\) depends on the cosine of \( t \)
• \(y(t)\) is often a constant if the ellipse lies on a specific plane
• \(z(t)\) varies with the sine of \( t \)

In the given exercise, the ellipse has a fixed \( y \)-coordinate because it is aligned parallel to the \( xz \)-plane and runs along \((0,1,-2)\). This fixed value of \( y \) helps maintain the ellipse's plane alignment as \( t \) varies.

The axes of an ellipse are crucial for understanding its geometry:- **Semi-Major Axis (a):** Half of the longest diameter of the ellipse, aligning with the direction where the ellipse extends the most.- **Semi-Minor Axis (b):** Half of the shortest diameter, perpendicular to the semi-major axis, representing the smallest span of the ellipse.For an ellipse described by diameters, computing these involves dividing the diameters by 2:

• The given major diameter of 3 translates to a semi-major axis of \( a = \frac{3}{2} = 1.5 \).
• A minor diameter of 2 turns into a semi-minor axis of \( b = \frac{2}{2} = 1 \).

These values define the scale of cosine and sine components in parameterization, indicating how much the ellipse stretches in each respective direction relative to its center.

Sometimes, the ellipse is not centered at the origin. This requires a shift in the coordinate system to correctly describe its placement. Such a transformation involves adjusting each coordinate to account for the ellipse’s center:For ellipses away from the origin, say centered at \((x_0, y_0, z_0)\), the parameterization adjusts as follows:\[(x(t), y(t), z(t)) = (x_0 + a \cos(t), y_0, z_0 + b \sin(t))\]In the exercise, the ellipse is centered at \((0,1,-2)\), requiring:

• Add 0 to the x-coordinate for horizontal alignment.
• The y-coordinate, 1, as it stays constant throughout.
• -2 added to the z-coordinate, shifting the center down on the z-axis.

This adjustment helps accurately place the ellipse in a non-origin location while maintaining its correct shape and orientation.