Cross-section (geometry)

In geometry and science, a cross section is the intersection of a body in three-dimensional space with a plane, or the analog in higher-dimensional space. Cutting an object into slices creates many parallel cross sections. A cross section of three-dimensional space that is parallel to two of the axes is a contour line; for example, if a plane cuts through mountains of a raised-relief map parallel to the ground, the result is a contour line in two-dimensional space showing points of equal elevation.

Conic sections – circles, ellipses, parabolas, and hyperbolas – are formed by cross-sections of a cone at various different angles, as seen in the diagram at left.

Any planar cross-section passing through the center of an ellipsoid forms an ellipse on its surface, which degenerates to a circle for sections perpendicular to a symmetry axis.

A cross-section of a cylinder is a circle if the cross-section is parallel to the cylinder's base, or an ellipse with non-zero eccentricity (see diagram at right) if it is neither parallel nor perpendicular to the base. If the cross-section is perpendicular to the base it consists of two parallel line segments (not shown) unless it is just tangent to the cylinder, in which case it is a single line segment.

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