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Quadratic

Quadric

Quartic


Quadratic

A quadratic is a second-degree polynomial. One may encounter quadratic expressions or quadratic equations. A quadratic expression f(x) is usually written as

f(x) = ax^2 + bx + c

where a, b, c are real constants and a is not 0. The graph of y = f(x) in 2-D Cartesian coordinates is a parabola. A quadratic equation is written as

f(x) = 0, i.e

ax^2 + bx + c = 0

The roots p and q of this equation are given by

p = (-b - sqrt(b^2 - 4ac)) / 2a
q = (-b + sqrt(b^2 - 4ac)) / 2a

The roots are real only when b^2 - 4ac is positive (or zero, in which case there is only one root). In terms of the graph of y = f(x), the roots are the values of x at which the parabolic graph intersects the x-axis. Quadric shapes such as circles, ellipses, parabolas, spheres, cylinders, hyperboloids etc all give quadratic equations when their intersections with a ray are considered. Therefore solving quadratic equations is a common function in a raytracer.

Quadric

A quadric is a locus in 3-dimensional space that can be represented in Cartesian coordinates as a polynomial equation in x, y and z of the second degree. It is the 3-D analog of a conic (circle, ellipse, parabola, hyperbola, pair of straight lines). The general quadric equation may be written as

Quadric equation

When this equation represents a non-empty locus, it can be reduced by translation and rotation to one of the following "canonical forms":

i. Ellipsoid (including sphere):   

ii. Hyperboloid of 1 sheet:   

iii. Hyperboloid of 2 sheets:   

iv. Elliptic paraboloid:   

v. Hyperbolic paraboloid:   

vi. Quadric cone:   

vii. Elliptic cylinder:   

viii. Hyperbolic cylinder:   

ix. Parabolic cylinder:   

x. Pair of non-parallel planes:   

xi. Pair of parallel planes:   

xii. Plane:     ,   i.e. x = 0

xiii. Line:     ,   i.e. x = y = 0

xiv. Point:     ,   i.e. x = y = z = 0

Forms i, ii, iii, iv and v are the "non-degenerate quadrics".

Quartic

A quartic is a polynomial of degree 4. Often, quartic refers to a shape described as a polynomial function of the coordinates (say x, y, z), such that the highest degree of any term is 4. An example of such a shape is a torus. A quartic equation is an equation of the form f(x) = 0, where f(x) is a quartic polynomial. The method of solving this equation is due to the Italian mathematician Ludovici Ferrari. It consists of adding the term (mx + b)^2 to both sides of the equation so as to convert them into perfect squares (m and b are calculated with some algebraic acrobatics followed by the solution of a cubic equation). We now equate the square roots of these quantities with the two possible combinations of positive/negative signs (same sign or different signs). This gives us two quadratic equations. Solving these equations gives us the 4 roots (real or complex, distinct or repeated) of the quartic equation.

Siddhartha Chaudhuri, 2002