Table of Contents
- 1 How do you find the intersection of two spheres?
- 2 How do you find the equation of a sphere when given two points?
- 3 What is the intersection of three spheres?
- 4 How do you find the volume of a sphere in spherical coordinates?
- 5 How do you calculate the intersection between a ray and a plane sphere triangle?
- 6 How do you find the volume of a solid sphere?
How do you find the intersection of two spheres?
The intersection curve of two sphere always degenerates into the absolute conic and a circle. Therefore, the real intersection of two spheres is a circle. The plane determined by this circle is perpendicular to the line connecting the centers of the spheres and this line passes through the center of this circle.
How do you find the equation of a sphere when given two points?
The general equation of a sphere is: (x – a)² + (y – b)² + (z – c)² = r², where (a, b, c) represents the center of the sphere, r represents the radius, and x, y, and z are the coordinates of the points on the surface of the sphere.
How do you find the intersection of a sphere and a plane?
The intersection of this sphere with the xy-plane is the set of points on the sphere whose z-coordinate is 0. Putting z = 0 into the equation, we have (x – 2)2 + (y + 6)2 = 9, z = 0 which represents a circle in the xy-plane with center (2, -6,0) and radius 3.
What is the volume of the sphere?
The formula for the volume of a sphere is V = 4/3 πr³.
What is the intersection of three spheres?
This Demonstration illustrates how trilateration can be done using the intersection of three spheres. Each pair of spheres either do not intersect or intersect in a point (when the spheres are tangent) or a circle….Spherical Cycloid.
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How do you find the volume of a sphere in spherical coordinates?
(1) V = 4 3 π r 3 , where is the radius. Note the use of the word ball as opposed to sphere; the latter denotes the infinitely thin shell, or, surface, of a perfectly round geometrical object in three-dimensional space. A surface has no volume, hence, we prefer to refer to it as a ball.
How do you find the volume of a sphere using spherical coordinates?
Volume formula in spherical coordinates
- V = ∫ ∫ ∫ B f ( x , y , z ) d V V=\int\int\int_Bf(x,y,z)\ dV V=∫∫∫Bf(x,y,z) dV.
- where B represents the solid sphere and d V dV dV can be defined in spherical coordinates as.
- d V = ρ 2 sin d ρ d θ d ϕ dV=\rho^2\sin\ d\rho\ d\theta\ d\phi dV=ρ2sin dρ dθ dϕ
What is the equation of a sphere in spherical coordinates?
A sphere that has the Cartesian equation x2 + y2 + z2 = c2 has the simple equation r = c in spherical coordinates.
How do you calculate the intersection between a ray and a plane sphere triangle?
Ray–Sphere Intersection
- Sphere: dot((P−C),(P−C))=r2.
- Ray: p(t)=A+tB.
- Combined: dot((A+tB−C),(A+tB−C))=r2.
How do you find the volume of a solid sphere?
What is the formula for volume?
Whereas the basic formula for the area of a rectangular shape is length × width, the basic formula for volume is length × width × height.