Spheres

Prerequisites : Coordinate Geometry I and Coordinate Geometry II

  • Spheres

A sphere is an idealized entity, defined as the locus of a point in 3D space, such that its distance from a fixed point remains constant. If (a,b,c) is the fixed point and r is the fixed distance, then by distance formula,

{(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2 \ \ \ \ (I)}

r is the radius and (a,b,c) is the center of the sphere.

sphere_1.jpg

 

  • General Form of Equation of Sphere

{x^2 + y^2 + z^2 + 2gx + 2fy + 2hz + d = 0}

Radius is \sqrt {g^2 +f^2 + h^2 -d} and center is (-g,-f,-h).

The terms above can be rearranged to get the equation similar to (I)

  • Diameter Form

sphere_diameter_form.jpg

 

If P_1 and P_2 are endpoints of diameter of a sphere, then its equation is given by

{(x-x_1)(x_1-x_2)+ (y-y_1)(y-y_2)+ (z-z_1)(z_1-z_2) = 0}

  • Externally Touching Spheres

 

spheres_externally_touching.jpg

The segment joining C_1 and C_2 is divided internally in the ratio r_1 : r_2

  • Internally Touching Spheres

spheres_internally_touching.jpg

 

The segment joining C_1 and C_2 gets divided externally in the ratio r_1 : r_2

  • Tangent Plane to a Sphere

Let (x_1,y_1,z_1) be a point on the sphere x^2 + y^2 + z^2 + 2gx + 2fy + 2hz + d = 0. The equation of plane which touches the sphere at (x_1,y_1,z_1) is given by

{xx_1 + yy_1 + zz_1 + g(x+x_1)+f(y+y_1)+h(z+z_1)+d = 0}

sphere_tangent_plane.jpg

 

It goes without saying that the length of perpendicular to the plane from the center of the circle is equal to the radius of the sphere.

  • Section of Sphere by Plane

When a sphere is sliced by a plane, the section obtained is a circle. If the plane passes through the center of the sphere, the circle is known as great circle. If the equation of sphere is S=0 and that of plane is P=0, then the equations together represent the circle.

sphere_section.jpg

  • Section of Sphere by Another Sphere

When 2 spheres intersect each other, the section obtained is a circle. If the equations of spheres are S_1 =0 and S_2 = 0, the equations together represent the circle.

The equation S_1 - S_2 =0 represents the plane in which the circle lies. The plane is known as radical plane. Radical plane is the locus of all those points, from which the lengths of tangents to the spheres are equal.

  • Family of Spheres

I) If S=0 and P=0 represent a circle, then, S+\lambda P = 0 represents a family of spheres passing through the circle.

II) If S_1 = 0 and S_2 = 0 represent a circle, then S_1 + \lambda S_2 = 0 represents a family of spheres passing through the circle.

  • Orthogonal Spheres

If S_1 =0 and S_2 =0 intersect such that the tangent planes T_1 = 0 and T_2 =0 at the point of contact are perpendicular to each other, then the spheres are said to be orthogonal.

sphere_orthogonal.jpg

 

Let the equations be

{S_1 = x^2 + y^2 + z^2 + 2g_1x + 2f_1y + 2h_1z + d_1 = 0}

and

{S_2 = x^2 + y^2 + z^2 + 2g_2x + 2f_2y + 2h_2z + d_2 = 0}

Then, the condition for orthogonality is

{2(g_1 g_2 + f_1 f_2 + h_1 h_2) = d_1 + d_2}

 

 

 

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