Cross section (geometry)

**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.https://en.wikipedia.org/wiki/Cross_section_(geometry)