Cross section (geometry)

Definition and Types of CrossSections: A crosssection is the intersection of a solid with a plane. The shape of a crosssection varies based on the cutting plane’s orientation. Crosssections 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 crosssections of polyhedrons, conic sections like circles and ellipses, and solid cylinders with disk or elliptic crosssections. Applications and Visualization of CrossSections: Cutting planes in computed axial tomography generate crosssections from xray data. Plane sections help visualize functions, derivatives, and partial derivatives of functions. Conditional density functions and isodensity 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. CrossSections in Related Subjects: Cavalieri’s principle states that equal crosssectional areas correspond to equal volumes. Conditional density functions and isodensity 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. Crosssections 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 crosssections and related topics.