Surfaces and Coordinates
Surfaces
Planes
A sheet on a 3D graph
A(x-x0)+B(y-y0)+C(z-z0)=0
(A, B, C)= Normal Vector
Different from Parametrized Curves, this normal vector is perpendicular to the plane in the equation
(x0, y0, z0)= a point on plane
Can be found by using techniques to find perpendiculars, such as cross product
Alternative:
x/a + y/b + z/c= 1
a, b, and c are the intercepts of their respective axis
Spheres and Ellipsoids
Sphere: (x-xc)^2 +(y-yc)^2 +(z-zc)^2=r^2
(xc, yc, zc) = the center of the sphere
Ellipsiod: x^2/a^2 +y^2/b^2 +z^2/c^2 =1
a, b, and c are intercepts of respective axis
Coordinates
Rectangular or Cartesian
(x, y, z)
Cylindrical
(r, theta, z)
r= sqrt(x^2 +y^2)
theta= arctan(y/x) (+ pi if x<0)
Spherical
(rho, theta, phi)
rho (p)
=sqrt(x^2+y^2+z^2)
theta
same as cylindrical
phi
=arccos(z/rho)
x
=rcos(theta)
=rho*sin(phi)cos(theta)
y
=rsin(theta)
=rho*sin(phi)cos(theta)
z
=rho*cos(phi)