Hyperboloid, Ellipsoid, Paraboloid | Mathematics for Grade 12 PDF Download

Introduction

• A Quadratic (order 2) polynomial defines a “quadric surface,” which is an algebraic surface. In R3 (common 3-dimensional Euclidean space), non-degenerate quadrics are classified as ellipsoids, paraboloids, or hyperboloids. 
• Most kinds of quadric, including degenerate examples, are represented in our collection. 

Curves of Degree 2

• Let’s start with the two-dimensional equivalent of quadric surfaces: degree 2 plane curves. R² is a quadratic form in two-dimensional Euclidean space that defines these (that is, they are defined on the zero set of a polynomial in x,y where the highest power involved is two, or the xy term has a non-zero coefficient). x²+y²-1=0, 
  For Example, is the formula for the unit circle.
• Because each of a circle, ellipse, parabola, and hyperbola can be realised as the intersection of a cone with a plane, degree two curves fall into four (non-degenerate) categories and are known as “conic sections.” 

Quadric Surface Definition

• Ellipsoids are defined by
  x²/a²+y²/b²+z²/c²=1
• Imposing a=b results in a “spheroid,” while imposing a=b=c results in a sphere. Ellipsoid shapes can be employed in architecture to create stunning structures.
• Elliptic and hyperbolic paraboloids are two types of paraboloids. 
  The term “elliptic paraboloid” refers to a paraboloid that has an elliptic shape.
  x²/a²+y²/a²-z=1
• The particular case where a=b is called a “circular paraboloid.” The formula for a “hyperbolic paraboloid” is:
  x²/a² – y²/a²-z=1
• Elliptic paraboloids are made by stretching the surface of an ellipsoid to infinity along one axis.
  An “elliptic hyperboloid” is the general form of a hyperboloid, as defined by:
  x²/a²+y²/b²−z²/c²=±1
• We get a hyperboloid “of one sheet” if the right-hand side is positive, and a hyperboloid “of two sheets” if the right-hand side is negative. The particular case where a=b yields a “circular hyperboloid,” as before. 
• Cones (elliptic and circular) and cylinders (elliptic, circular, parabolic, and hyperbolic) are examples of degenerate quadric surfaces. Cones are ruled surfaces that can be obtained by applying a limiting process to hyperboloids, taking a,b,c to infinity.

Determining Ellipsoids and Conics

• The position of five distinct points lying on any conic section can be used to determine its uniqueness. The points must be in general position – no more than three of them can be collinear (all lying on one line). 
• As a result, there is a unique conic crossing through five such sites in R². As elegant as such constructs may appear, declaring five points is not a very practical way of identifying a conic (the following construction is much more complex than solving a degree 2 polynomial). 
• The major and minor axes dictate the shape of an ellipse, which can be made by hand with pins and string.
• The axes of an ellipsoid, like an ellipse, determine its shape. However, an ellipsoid has three main axes, which are indicated in red here. The ellipsoid is uniquely determined by the intersection of an ellipsoid with any three planes, and any such intersection with a plane is an ellipse (or a circle, which is a special case of an ellipse). Once again, the construction is simply too intricate for this characterisation to be effective for describing ellipsoids.
• Using linear algebra, there is a much more elegant technique to determine the type of a conic or quadric. Any ellipsoid or hyperboloid has a defining equation of the form x^TAx=1, where x=(x,y,z)  and A is a real symmetric matrix (equal to its transpose). 
• We may discover a rotation matrix P, satisfying PP^T=P^TP=I, the identity matrix, such that PAP^T=D, where the matrix D is diagonal with the eigenvalues of A as its diagonal elements, according to a well-known theorem in linear algebra. 
• The primary axes of the quadric are then A’s eigenvectors. We can compose x^T Ax=x^TP^TPAP^TPx=X^TDX=1, in which X=Px. 
• The coordinate axes are the primary axes of the quadric in the rotational coordinates X, and the signs of the eigenvalues (diagonal elements of D) determine the type. 
• It’s an ellipsoid if they’re all positive; two positive and one negative equals a one-sheet hyperboloid; one positive and two negative equals a two-sheet hyperboloid. 
• We have a degenerate case when one or more eigenvalues vanish, such as a paraboloid, cylinder, or even a couple of planes.