FORMULAS from SOLID ANALYTIC GEO
FORMULAS from SOLID ANALYTIC GEOMETRY
d = √(x2
- x1)2 + (y2 - y1)2 + (z2
- z1)2
DIRECTION COSINES OF LINE JOINING POINTS P1(x1,y1,z1)
AND P2(x2,y2,z2)
l = cosα = (x2 - x1)/d, m = cosβ = (y2 - y1)/d,
n = cosγ = (z2 - z1)/d
where α,β,γ are the angles which line P1P2 makes with the
positive x, y, z axes respectively and d is given by the figure above.
RELATIONSHIP BETWEEN DIRECTION COSINES
cos2α + cos2β + cos2γ = 1 or l2 + m2
+ n2 = 1
DIRECTION NUMBERS
Number L, M, N which are proportional to the direction cosines l, m, n are
called direction numbers. The relationship between them is given by
l = L/(√L2
+ M2 + N 2, m = M/(√L2
+ M2 + N 2, n = N/(√L2
+ M2 + N 2
EQUATIONS OF LINE JOINING P1(x1,y1,z1)
AND P2(x2,y2,z2) IN STANDARD FORM
(x - x1)/(x2 - x1) = (y - y1)/(y2
- y1) = (z - z1)/(z2 - z1) or (x - x1)/l
= (y - y1)/m = (z - z1)/n
These are also valid if l, m, n are replaced by L, M, N respectively.
EQUATIONS OF LINE JOINING P1(x1,y1,z1)
AND P2(x2,y2,z2) IN PARAMETRIC FORM
x = x1 + lt, y = y1 + mt, z = z1 + nt
These are also valid if l, m, n are replaced by L, M, N respectively.
ANGLE φ BETWEEN TWO LINES WITH DIRECTION COSINES l1, m1,
n1 AND l2, m2, n2
cosφ = l1l2 + m1m2 + n1n2
GENERAL EQUATION OF A PLANE
Ax + By + Cz + D = 0 [A, B, C, D are constants]
EQUATION OF PLANE PASSING THROUGH POINTS (x1,y1,z1),
(x2,y2,z2), (x3,y3,z3)
or
EQUATION OF A PLANE IN INTERCEPT FORM
x/a + y/b + z/c = 1
where a, b, c are the intercepts on the x, y, z axes respectively.
EQUATION OF LINE THROUGH (x0,y0,z0) AND
PERPENDICULAR TO PLANE Ax + By + Cz + D = 0
(x - x0)/A = (y - y0)/B = (z - z0)/C or x = x0
+ At, y = y0 + Bt, z = z0 + Ct
Note that the direction numbers for a line perpendicular to a plane Ax + By + Cz
+ D = 0 are A, B, C.
DISTANCE FROM POINT (x0,y0,z0) TO A PLANE
Ax + By + Cz + D = 0.
(Ax0 + By0 + Cz0 + D)/± √A2
+ B2 + C2
where the sign is chosen so that the distance is nonnegative.
NORMAL FORM FOR EQUATION OF PLANE
xcosα + ycosβ + zcosγ = p
where p = perpendicular distance from O to plane at P and α, β, γ are angles
between OP and positive x, y, z axes.
TRANSFORMATION OF COORDINATES INVOLVING PURE TRANSLATION
where (x, y, z) are old coordinates [i.e. coordinates relative to xyz system],(x',
y', z') are new coordinates [relative to the x'y'z' system] and (x0,y0,z0)
are coordinates of the new origin O' relative to the old xyz coordinate system.
TRANSFORMATION OF COORDINATES INVOLVING PURE ROTATION
where the origins of the xyz and x'y'z' systems are the same and l1,m1,n1;
l2,m2,n2; l3,m3,n2
are the direction cosines of the x', y', z' axes relative to the x, y, z axes
respectively.
TRANSFORMATION OF COORDINATES INVOLVING TRANSLATION AND ROTATION
where the O' of x'y'z' system has coordinates (x0,y0,z0)
relative to the xyz system and l1,m1,n1; l2,m2,n2;
l3,m3,n2 are the direction cosines of the x' ,
y', z' axes relative to the x, y, z axes respectively.
CYLINDRICAL COORDINATES (r, θ, z)
A point P can be located by cylindrical coordinates (r, θ, z) as well as
rectangular coordinates (x, y, z).
The transformation between these coordinates is
SPHERICAL COORDINATES (r, θ, φ)
A point P can be located by spherical coordinates (r, θ, φ) as well as
rectangular coordinates (x, y, z).
The transformation between those coordinates is
EQUATION OF A SPHERE IN RECTANGULAR COORDINATES
(x - x0)2 + (y - y0)2 + (z - z0)2
= R2
where the sphere has center (x0,y0,z0) and
radius R.
EQUATION OF SPHERE IN CYLINDRICAL COORDINATES
r2 - 2r0r(θ - θ0) + r02 +
(z - z0)2 = R2
where the sphere nas center (r0,θ0,z0) in
cylindrical coordinates and radius R.
If the center is at the origin the equation is:
r2 + z2 = R2
EQUATION OF SPHERE IN SPHERICAL COORDINATES
r2 + r02 - 2r0rsinθsinθ0cos(φ
- φ0) = R2
where the sphere has center (r0,θ0,φ0) in
spherical coordinates and radius R.
If the center is at the origin the equation is
r = R.
EQUATION OF ELLIPSOID WITH CENTER (x0,y0,z0)
AND SEMI-AXES a,b,c
(x - x0)2/a2 + (y - y0)2/b2
+ (z - z0)2/c2 = 1
ELLIPTIC CYLINDER WITH AXIS AS z AXIS
x2/a2 + y2/b2 = 1
where a, b are semi axes of elliptic cross section.
If b = a it becomes a circular cylinder of radius a.
ELLIPTIC CONE WITH AXIS AS z AXIS
x2/a2 + y2/b2 = z2/c2
HYPERBOLOID OF ONE SHEET
x2/a2 + y2/b2 - z2/c2
= 1.
HYPERBOLOID OF TWO SHEETs
x2/a2 - y2/b2 - z2/c2
= 1.
ELLIPTIC PARABOLOID
x2/a2 + y2/b2 = z/c
HYPERBOLIC PARABOLOID
x2/a2 - y2/b2 = z/c
Note orientation of axes in the figure.
