kegelform

Kegelform, or cone form, refers to a geometric shape consisting of all straight lines that pass through a fixed point, the apex, and intersect a fixed base curve or surface. The resulting set can be a solid or a surface, depending on whether the base is a region or a curve. The simplest and most familiar example is the right circular cone, produced when the base is a circle lying in a plane perpendicular to the axis through the apex.

Common types and measurements

- Right circular cone: base radius r, height h (the perpendicular distance from apex to the base plane). The slant height is l = sqrt(r^2 + h^2). The lateral surface area is π r l, and the total surface area is π r (r + l). The volume is V = (1/3) π r^2 h.

- Oblique cone: the apex is offset from the base center, producing a leaning form. If the base remains a circle, the volume formula V = (1/3) base_area × height still applies, but the geometry of the surface becomes more intricate.

- General cone: defined by a fixed apex and a base curve or surface; every cross-section by a plane parallel to the base is similar to the base (in the right circular case) and yields consistent scaling relations.

Mathematical descriptions

- In Cartesian coordinates, a right circular cone with apex at the origin and axis along z can be described by x^2 + y^2 = (tan θ)^2 z^2, where θ is the half-angle at the apex.

- In algebraic and projective geometry, a cone is the union of all lines through a vertex meeting a fixed base variety; cones often serve as basic examples of singular varieties.

Applications and context

Kegelforms appear in architecture, optics, computer graphics, and modeling where a linear spread from a point is exploited. They also provide simple models for volume and surface-area problems and appear in more abstract mathematical contexts as cones over base objects.