91Ó°ÊÓ

To understand the geometry of a 3D surface like an ellipsoid, one effective method is to examine its traces. Traces are the intersections of the surface with coordinate planes. These traces give us a sense of the surface's shape in different dimensions.

By setting one of the coordinates to zero in the ellipsoid equation, we can find three distinct traces:

• Setting \( z = 0 \), we obtain an equation for the ellipse in the XY-plane: \( \frac{x^{2}}{9} + \frac{y^{2}}{25} = 1 \).
• For the YZ-plane, by setting \( x = 0 \), the trace is \( \frac{y^{2}}{25} + \frac{z^{2}}{4} = 1 \).
• Lastly, setting \( y = 0 \) gives the trace in the XZ-plane as \( \frac{x^{2}}{9} + \frac{z^{2}}{4} = 1 \).

These traces show that each cross-section within these planes forms an ellipse. This is crucial because the shape and size of these ellipses inform us about how the ellipsoid stretches or compresses along each axis. By analyzing these traces, we gain better insight into the structure and orientation of the whole surface.

An ellipsoid is a three-dimensional shape defined by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \). This formula represents the standard form of an ellipsoid, where \( a, b, \) and \( c \) are the semi-axis lengths.

The given exercise utilizes this form. The equation \( \frac{x^{2}}{9} + \frac{y^{2}}{25} + \frac{z^{2}}{4} = 1 \) clearly shows the property of an ellipsoid because each term represents one squared axis length denominator:

• \( a^2 = 9 \) indicates a semi-major axis length of \( a = 3 \) along the x-axis.
• \( b^2 = 25 \) gives \( b = 5 \) along the y-axis.
• \( c^2 = 4 \) means \( c = 2 \) on the z-axis.

Understanding these terms helps us see how the ellipsoid stretches along each axis. Differing values of \( a, b, \) and \( c \) give each ellipsoid unique proportions and orientations, defining its overall shape and volume. This knowledge is essential for interpreting the spatial properties of an ellipsoid.

Cross-sections provide a crucial means to visualize and comprehend ellipsoids. These sections are created when the ellipsoid is sliced by a plane, and in standard presentations, these slices are often aligned with the coordinate planes.

In our exercise, examining the cross-sections helps confirm the ellipsoid's form. We've seen sections that are ellipses lined up along different planes:

• In the XY-plane, we have a cross-section with an ellipse defined by \( \frac{x^{2}}{9} + \frac{y^{2}}{25} = 1 \). This section is wider along the y-axis due to the \( b = 5 \) semi-major axis.
• The YZ-plane yields an ellipse \( \frac{y^{2}}{25} + \frac{z^{2}}{4} = 1 \), emphasizing the length along the y-axis.
• The XZ-plane shows an ellipse described by \( \frac{x^{2}}{9} + \frac{z^{2}}{4} = 1 \), where the primary stretch is along the x-axis with \( a = 3 \).

These cross-sections demonstrate how the ellipsoid extends in space. The fundamental idea is that each plane cut reveals a two-dimensional representation of this three-dimensional object, showing varying widths and heights that help us understand the full geometry of the ellipsoid. This insight gives us the ability to envision and work with ellipsoids in complex spatial scenarios.