Term: Cross section (geometry)

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**Definition and Types of Cross-Sections:**
– A cross-section is the intersection of a solid with a plane.
– The shape of a cross-section varies based on the cutting plane’s orientation.
– Cross-sections of a ball are disks, while those of a cube depend on the cutting plane’s orientation.
– Plane sections are curves where a plane intersects a surface.
– Mathematical examples include cross-sections of polyhedrons, conic sections like circles and ellipses, and solid cylinders with disk or elliptic cross-sections.

**Applications and Visualization of Cross-Sections:**
– Cutting planes in computed axial tomography generate cross-sections from x-ray data.
– Plane sections help visualize functions, derivatives, and partial derivatives of functions.
– Conditional density functions and iso-density contours in probability, production functions, and utility functions in economics can be represented by plane sections.
– Isoquants and indifference curves are examples of plane sections in economics.

**Cross-Sections in Related Subjects:**
– Cavalieri’s principle states that equal cross-sectional areas correspond to equal volumes.
– Conditional density functions and iso-density contours for the normal distribution are examples of plane sections.
– Production functions, cardinal or ordinal utility functions, isoquants, and indifference curves in economics can be visualized using plane sections.

**Descriptive Geometry and Graphical Projection:**
– Descriptive geometry involves the study of geometric principles through drawings and helps visualize 3D objects in 2D space.
– Graphical projection is a technique to represent 3D objects on 2D surfaces, essential in technical drawings and drafting.
– Cross-sections are crucial in both descriptive geometry and graphical projection for clarity and conveying spatial relationships.

**Additional Concepts and Resources:**
– Section lining, representation of materials, profile gauge, and secant plane are related concepts.
– Descriptive geometry includes orthographic projections.
– Graphical projection types include isometric, oblique, and perspective projections.
– Resources like Wikimedia Commons and references like Swokowski and Albert provide further information on cross-sections and related topics.

In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross-sections. The boundary of a cross-section in three-dimensional space that is parallel to two of the axes, that is, parallel to the plane determined by these axes, is sometimes referred to as 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 on the surface of the mountains of equal elevation.

A cross-section view of a compression seal

In technical drawing a cross-section, being a projection of an object onto a plane that intersects it, is a common tool used to depict the internal arrangement of a 3-dimensional object in two dimensions. It is traditionally crosshatched with the style of crosshatching often indicating the types of materials being used.

With computed axial tomography, computers can construct cross-sections from x-ray data.

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