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Three Dimensions

The most commonly used coordinate system in three dimensions is the Cartesian or rectangular coordinate system described below. Two other systems seen occasionally are the cylindrical coordinate system and the spherical coordinate system.


                                              ^ y
                                              |
                                              |     x
                                             2* - - - - -_*
                                         _ -  |      _ -  .
                                   z _ -      | z_ -      .         _,|'
                                 _ -         _|-          . y   _,|'  -6
                             _ -    x    _ - 1+           . _,|'
                           * - - - - - *      |         _,|'  -4
                           .   P(x,y,z).      |     _,|'  .
                           .           .      | _,|'  -2  .
                       ----.+-----+----.+-----+-----+-----*-----+-----> x
                           -3    -2    -1,|' 0|     1_ -  2     3
                         y .        _,|'      |  _ -
                           .    _,|'   2     _|-  z
                           ._,|'       . _ --1+
                      z _,|* - 4 - - - *      |
                      <'         x            |
                                              |
                                            -2+
                                              |
                                              |

[
Back to Analytic Geometry Formula Contents]

Three dimensions: Points

A point is specified by an ordered triple of numbers called its coordinates. Let the coordinates of Pi be (xi,yi,zi), for i = 1, 2, 3, and 4.

The distance from P1 to P2 is

d = sqrt[(x1-x2)2+ (y1-y2)2+ (z1-z2)2].

The coordinates of the point dividing the line segment P1P2 in the ratio r/s are:

([r x2+s x1]/[r+s], [r y2+s y1]/[r+s], [r z2+s z1]/[r+s]).

As a special case, when r = s, the midpoint of the line segment has coordinates

([x2+x1]/2, [y2+y1]/2, [z2+z1]/2).

P1, P2, and P3 are collinear if and only if

(x2-x1)/(x3-x1) = (y2-y1)/(y3-y1) = (z2-z1)/(z3-z1)

P1, P2, P3, and P4 are coplanar if and only if the determinant


 x1   y1   z1   1 
 = 0.
 x2   y2   z2   1 
 x3   y3   z3   1 
 x4   y4   z4   1 


[ Back to Analytic Geometry Formula Contents]

Three dimensions: Directions

A direction is determined by an ordered triple of three numbers (a,b,c), not all zero, called direction numbers. The direction corresponds to all lines parallel to the line through the origin (0,0,0) and the point (a,b,c). The direction numbers (a,b,c) and the direction numbers (ra,rb,rc) determine the same direction, for any nonzero r.

We pick an r in the following way.

|r| = 1/sqrt(a2+b2+c2).

If a is nonzero, the sign of r is the same as the sign of a. If a = 0 and b is nonzero, the sign of r is the same as the sign of b. If a = b = 0, the sign of r is the same as the sign of c. This means that the first nonzero number in (ra,rb,rc) is positive.

With this choice of r, the direction numbers (ra,rb,rc) are just the cosines of the angles that the line makes with the positive x-, y-, and z-axes. These angles alpha, beta, and gamma, respectively, are called the direction angles, and their cosines (cos[alpha],cos[beta],cos[gamma]) are called the direction cosines of that direction. They satisfy

cos2[alpha] + cos2[beta] + cos2[gamma] = 1.

Two directions are parallel if and only if they have the same direction angles if and only if they have the same direction cosines.

Two directions are perpendicular if and only if

cos(alpha1)cos(alpha2) + cos(beta1)cos(beta2) + cos(gamma1)cos(gamma2) = 0.

The angle between two directions is given by

arccos[cos(alpha1)cos(alpha2) + cos(beta1)cos(beta2) + cos(gamma1)cos(gamma2)].

The direction perpendicular to two given directions (a1,b1,c1) and (a2,b2,c2) has direction numbers (a3,b3,c3), where

a3 = b1c2 - c1b2,
b3 = c1a2 - a1c2,
c3 = a1b2 - b1a2.

Three directions are parallel to a common plane if and only if


 a1   b1   c1 
= 0.
 a2   b2   c2 
 a3   b3   c3 


[ Back to Analytic Geometry Formula Contents]

Three dimensions: Lines

A line is the set of points whose coordinates (x,y,z) satisfy two equations of the first degree in x, y, and z. (That means it is the intersection of two planes.) The equations of a line can take several forms:

  • Point direction form:

    (x-x1)/a = (y-y1)/b = (z-z1)/c.

    If any of the denominators is zero (say the first), remove that part of the above equations and replace it with an equation of the form x = x1.

  • Two point form:

    (x-x1)/(x2-x1) = (y-y1)/(y2-y1) = (z-z1)/(z2-z1).

    If any of the denominators is zero (say the first), remove that part of the above equations and replace it with an equation of the form x = x1.

  • Parametric form:

    x = x1 + t cos(alpha),
    y = y1 + t cos(beta),
    z = z1 + t cos(gamma),

    where the parameter t is any real number.

The length of the projection of line segment P1P2 on any line in the direction (a,b,c) is

[(x2-x1)a + (y2-y1)b + (z2-z1)c]/sqrt[a2+b2+c2].

The distance from P2 to the line through P1 in the direction (a,b,c) is

sqrt([c(y2-y1)-b(z2-z1)]2 + [a(z2-z1)-c(x2-x1)]2 + [b(x2-x1)-a(y2-y1)]2) / sqrt(a2+b2+c2).

The distance between two lines, one through P1 in direction (a1,b1,c1), and the other through P2 in direction (a2,b2,c2), is given by:


 x2–x1   y2–y1   z2–z1 
 a1   b1   c1 
 a2   b2   c2 
±
.
sqrt(

 b1   c1 
2
 b2   c2 
 + 

 c1   a1 
2
 c2   a2 
 + 

 a1   b1 
2
 a2   b2 
)


As a corollary, the two lines intersect if and only if


 x2–x1   y2–y1   z2–z1 
 = 0.
 a1   b1   c1 
 a2   b2   c2 


[ Back to Analytic Geometry Formula Contents]

Three dimensions: Planes

A plane is the set of all points whose coordinates (x,y,z) satisfy a linear equation in x, y, and z. The equation of a plane can take any of the following forms.

  • Point direction form:

    a(x-x1) + b(y-y1) + c(z-z1) = 0.

    The point P1 lies in the plane, and the direction (a,b,c) is perpendicular (or normal) to the plane.

  • General form:

    Ax + By + Cz + D = 0.

    The normal direction has direction numbers (A,B,C).

  • Intercept form:

    x/a + y/b + z/c = 1,

    where none of a, b, or c is zero. The plane passes through the points (a,0,0), (0,b,0), and (0,0,c), which are called the x-, y-, and z-intercepts, respectively.

  • Three point form:


     x–x3   y–y3   z–z3 
     = 0,
     x1–x3   y1–y3   z1–z3 
     x2–x3   y2–y3   z2–z3 

    or else


     x   y   z   1 
     = 0.
     x1   y1   z1   1 
     x2   y2   z2   1 
     x3   y3   z3   1 

  • Normal form:

    x cos(alpha) + y cos(beta) + z cos(gamma) = p,

    where p is the perpendicular distance from the origin to the plane, and (cos[alpha],cos[beta],cos[gamma]) are the direction cosines of any line normal to the plane.

  • Parametric form:

    x = x1 + a1s + a2t,
    y = y1 + b1s + b2t,
    z = z1 + c1s + c2t,

    where the parameters s and t are any real numbers. The directions (a1,b1,c1) and (a2,b2,c2) are parallel to the plane.

The dihedral angle between two planes is equal to the angle between their normal directions. If the planes are given by

A1x + B1y + C1z + D1 = 0,
A2x + B2y + C2z + D2 = 0,

then the dihedral angle between them is

arccos([A1A2 + B1B2 + C1C2]/[sqrt(A12 + B12 + C12)sqrt(A22 + B22+C22)]).

As a corollary, the planes are parallel if and only if

A1/A2 = B1/B2 = C1/C2.

As another corollary, the planes are perpendicular if and only if

A1A2 + B1B2 + C1C2 = 0.

The equation of a plane through P1 and parallel to directions (a1,b1,c1) and (a2,b2,c2) has equation


 x–x1   y–y1   z–z1 
 = 0.
 a1   b1   c1 
 a2   b2   c2 

The equation of a plane through P1 and P2, and parallel to direction (a,b,c), has equation


 x–x1   y–y1   z–z1 
 = 0.
 x2–x1   y2–y1   z2–z1 
 a   b   c 

The distance of P1 from the plane Ax + By + Cz + D = 0 is

(Ax1 + By1 + Cz1 + D)/sqrt(A2+B2+C2).

The angle between a plane and a line is the complement of the angle between the line and an intersecting line perpendicular to the plane.

The intersection point of the line with equations

x = x1 + at,
y = y1 + bt,
z = z1 + ct,

and the plane with equation

Ax + By + Cz + D = 0,

has parameter value

t = -(Ax1+By1+Cz1+D)/(aA+bB+cC).

The intersection of two planes

A1x + B1y + C1z + D1 = 0,
A2x + B2y + C2z + D2 = 0,

is the line

x = x1 + at,
y = y1 + bt,
z = z1 + ct, or
(x-x1)/a = (y-y1)/b = (z-z1)/c,

where

a = 
 B1   C1 
,
 B2   C2 

b = 
 C1   A1 
,
 C2   A2 

c = 
 A1   B1 
,
 A2   B2 

b
 D1   C1 
 – c
 D1   B1 
 D2   C2   D2   B2 
x1
,
a2+b2+c2

c
 D1   A1 
 – a
 D1   C1 
 D2   A2   D2   C2 
y1
,
a2+b2+c2

a
 D1   B1 
 – b
 D1   A1 
 D2   B2   D2   A2 
z1
,
a2+b2+c2



If a = b = c = 0, then the planes are parallel.


[ Back to Analytic Geometry Formula Contents]

Three dimensions: Triangles

The area of a triangle with vertices P1, P2, and P3 is given by

K = (1/2)sqrt(

 y1   z1   1 
2
 y2   z2   1 
 y3   z3   1 
 + 

 z1   x1   1 
2
 z2   x2   1 
 z3   x3   1 
 + 

 x1   y1   1 
2
 x2   y2   1 
 x3   y3   1 
).


[ Back to Analytic Geometry Formula Contents]

Three dimensions: Tetrahedra

The volume of a tetrahedron with vertices P1, P2, P3, and P4 is given by

V = (1/6)
 x1   y1   z1   1 
,
 x2   y2   z2   1 
 x3   y3   z3   1 
 x4   y4   z4   1 

or by

V = (1/6)
 x1–x4   y1–y4   z1–z4 
.
 x2–x4   y2–y4   z2–z4 
 x3–x4   y3–y4   z3–z4 

The centroid of the tetrahedron has coordinates

x = (x1+x2+x3+x4)/4,
y = (y1+y2+y3+y4)/4,
z = (z1+z2+z3+z4)/4.

Easy generalizations of the formulas for the incenter, circumcenter, and orthocenter of a triangle hold for the tetrahedron.


[ Back to Analytic Geometry Formula Contents]

Three dimensions: General Quadratic Equations and Quadric Surfaces

A general quadratic equation can be put into the following form:

ax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy + 2px + 2qy + 2rz + d = 0.

If we have an equation of this kind, it can represent one of seventeen different kinds of surface, called quadric surfaces. Which kind depends on the values of the following four quantities:

    [ a  h  g ]       [ a  h  g  p ]
e = [ h  b  f ],  E = [ h  b  f  q ],
    [ g  f  c ]       [ g  f  c  r ]
                      [ p  q  r  d ]

rho3 = rank(e), rho4 = rank(E), Delta = det(E), and k1, k2, k3, the roots of det(xI-e) = 0.

The cases are as follows:

Case   rho3   rho4   Delta   k's same sign?   Quadric Surface                Standard Form

  1     3      4       -          yes         Real ellipsoid                 x2/a2+y2/b2+z2/c2 = 1
  2     3      4       +          yes         Imaginary ellipsoid            x2/a2+y2/b2+z2/c2 = -1
  3     3      4       +          no          Hyperboloid of 1 sheet         x2/a2+y2/b2-z2/c2 = 1
  4     3      4       -          no          Hyperboloid of 2 sheets        x2/a2+y2/b2-z2/c2 = -1
  5     3      3                  no          Real quadric cone              x2/a2+y2/b2-z2/c2 = 0
  6     3      3                  yes         Imaginary quadric cone         x2/a2+y2/b2+z2/c2 = 0
  7     2      4       -          yes         Elliptic paraboloid            x2/a2+y2/b2-z = 0
  8     2      4       +          no          Hyperbolic paraboloid          x2/a2-y2/b2-z = 0
  9     2      3                  yes         Real elliptic cylinder         x2/a2+y2/b2 = 1
 10     2      3                  yes         Imaginary elliptic cylinder    x2/a2+y2/b2 = -1
 11     2      3                  no          Hyperbolic cylinder            x2/a2-y2/b2 = 1
 12     2      2                  no          Real intersecting planes       x2/a2-y2/b2 = 0
 13     2      2                  yes         Imaginary intersecting planes  x2/a2+y2/b2 = 0
 14     1      3                              Parabolic cylinder             x2/a2-y = 0
 15     1      2                              Real parallel planes           x2/a2 = 1
 16     1      2                              Imaginary parallel planes      x2/a2 = -1
 17     1      1                              Coincident planes              x2 = 0
The equation can be put into standard form by making a rotation to remove the yz-, zx-, and xy-terms, completing the square(s), and then making a translation to move the center to the origin.


[ Back to Analytic Geometry Formula Contents]

Three dimensions: Spheres

A sphere is an ellipsoid with a = b = c = r. Its equation can take any of the following forms:

  • Center radius form:

    (x-x1)2 + (y-y1)2 + (z-z1)2 = r2,

    where the center is P1 and the radius is r.

  • General form:

    Ax2 + Ay2 + Az2 + Dx + Ey + Fz + M = 0,
    x2 + y2 + z2 + 2dx + 2ey + 2fz + m = 0,

    where A is nonzero. The center is (-d,-e,-f) = (-D/2A,-E/2A,-F/2A), and the radius is r = sqrt(d2+e2+f2-m).

  • Diameter form:

    (x-x1)(x-x2) + (y-y1)(y-y2) + (z-z1)(z-z2) = 0.

    where P1 and P2 are the ends of a diameter.

  • Four point form:


     x2+y2+z2   x   y   z   1 
     = 0.
     x12+y12+z12   x1   y1   z1   1 
     x22+y22+z22   x2   y2   z2   1 
     x32+y32+z32   x3   y3   z3   1 
     x42+y42+z42   x4   y4   z4   1 


Compiled by Robert L. Ward.

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