In three-dimensional space, two points uniquely determine a line and three non-colinear points uniquely determine a plane. More generally, a locally one-dimensional space is called a curve and a locally two-dimensional space is called a surface. Any single equation involving \(x\) and \(y\) and \(z\) analytically defines a surfaces in three-dimensional space, whereas curves, specifically lines, can’t be described by a single equation; curves must be described either parametrically or as the intersection of two surfaces.
The template equation for a plane in space passing through the point \((x_0, y_0, z_0)\) is either \( {A(x - x_0) + B(y - y_0) + C(z - z_0) = 0} \) or \( {Ax + By + Cz = D\,,} \) where the coefficients \(A\) and \(B\) and \(C\) correspond to the “slopes” of the plane in various directions.
A line is defined as the intersection of two planes, but it’s more convenient to describe it parametrically. A line through point \((x_0, y_0, z_0)\) with “direction” determined by coefficients \(A\) and \(B\) and \(C\) is given by the equations
A cylinder in general is a surface that results from translating a planar curve along a line orthogonal to the curve. The locus of a single point from the curve along the line is referred to as a ruling, and a single “slice” along a ruling will be a cross-section, more generally called a trace, congruent to the curve. A “cylinder” as you knew it previously is a specific example we’ll now refer to as a right circular cylinder.
A quadric surface is a surface with corresponding equation containing terms with at most degree two, e.g. \(x^2\) or \(z^2\) or \(yz.\) One specific example: a sphere of radius \(r\) centered at \((x_0,y_0,z_0)\) has equation \( r^2 = (x-x_0)^2+(y-y_0)^2+(z-z_0)^2\,. \)