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Surfaces and Coordinates - Coggle Diagram
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
Can be found by using techniques to find perpendiculars, such as cross product
(x0, y0, z0)= a point on plane
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)
x
=rcos(theta)
=rho*sin(phi)cos(theta)
y
=rsin(theta)
=rho*sin(phi)cos(theta)
z
=rho*cos(phi)
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)