A literature review of quadric surface theory

Quadric surfaces are a type of surface defined by a degree-two polynomial equation. They are commonly used in computer graphics, computer vision, and engineering applications. This paper presents a literature review of quadric surface theory, covering topics such as their mathematical definition, properties and applications. The mathematical definition of quadric surfaces is based on the concept of homogeneous coordinates. A point in three-dimensional space can be represented as a vector with four components: x, y, z and w. A quadric surface is then defined as the set of points satisfying an equation given by ax^2 + by^2 + cz^2 + dw^2 = 0. This equation can be written in terms of Cartesian coordinates by introducing the concept of homogeneous coordinates: x/w, y/w and z/w. The resulting equation is called the canonical form of the quadric surface and it has six parameters that define its shape: a, b, c, d (the coefficients) and u0 , v0 (the translation parameters). The properties of quadric surfaces depend on their coefficients. In particular, if all coefficients are positive then the surface is convex; if one coefficient is negative then it is concave; if two coefficients are negative then it has two connected components; if three or more coefficients are negative then it has multiple connected components or even no solution at all (in which case it is called degenerate). The type of curvature also depends on the coefficients: when all coefficients are equal to 1 (a=b=c=d=1) then the surface has constant curvature; otherwise it will have varying curvature along different directions. Quadric surfaces have many applications in computer graphics and engineering due to their simple mathematical definition and wide range of shapes they can represent. In computer graphics they are often used to model smooth curved objects such as spheres or ellipsoids. They can also be used for terrain modeling or for representing implicit functions such as distance fields or level sets. In engineering they have been used for robot path planning and collision detection algorithms due to their ability to represent complex shapes with simple equations. In conclusion, this paper presented an overview of quadric surface theory including its mathematical definition along with its properties and applications in computer graphics and engineering domains. Quadrics offer an efficient way to represent curved objects using simple equations which makes them suitable for many practical tasks such as terrain modeling or robot navigation algorithms