PROJECTIVE CONE Meaning and
Definition
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A projective cone is a geometric object formed by taking a base shape or figure and extending it infinitely in a cone-like manner, while simultaneously projecting all points from a fixed vertex outside the base. This concept is often used in projective geometry and has applications in various fields, including computer graphics, architecture, and mathematics. The resulting cone is often called the projective cone of the base shape.
In mathematics, a projective cone can be described as a surface or solid obtained by joining all lines passing through a fixed point called the vertex, to points on a fixed curve. The base curve can be any closed or open curve in space while the vertex lies outside the curve. For example, a projective cone can be formed by connecting all points on a given circle to a single point outside the circle.
The property that makes projective cones unique is the projection of points. Any point on the base shape is projected to a corresponding point on the surface of the cone through the vertex. This projection preserves the collinearity of points lying on a straight line in the base, resulting in a natural extension of the base shape to the surface of the cone.
Projective cones are studied extensively in projective geometry, as they allow for the exploration of properties specific to projective transformations and mappings. They are also utilized in computer graphics and architectural design, where they can be used to create visually appealing and aesthetically pleasing structures and objects.
Etymology of PROJECTIVE CONE
The term "projective cone" has its roots in mathematics and geometry. To understand its etymology, we need to break it down into its components: "projective" and "cone".
The word "projective" comes from the Latin word "proiectus", which means "thrown forth" or "thrown forward". It is derived from the verb "proicere", meaning "to throw forward" or "to extend outward". The term "projective" is often used in mathematics to describe properties of mathematical objects that remain invariant under projection or mapping.
The term "cone" comes from the Latin word "conus", which is derived from the Greek word "koneus" or "kōnos". It refers to a solid geometric figure with a flat base and a curved surface that tapers to a point or vertex. Cones are named for their resemblance to the shape of a conifer tree.
