Intersection of a Sphere and a Plane: Deriving the Equation of the Intersection Circle

Understanding the relationship between geometric shapes in space, such as spheres and planes, is crucial in various fields including mathematics, physics, and engineering. One specific scenario is the intersection of a sphere and a plane, which results in a circle. This article will focus on the process of deriving the equation of the circle formed by the intersection of the sphere x2 y2 z2 1 and the plane x y z 1. This problem challenges us to leverage both algebraic and geometric methods, offering insight into complex spatial relationships.

Step-by-Step Derivation

Let's begin by defining the given sphere and plane:

Sphere

The given sphere is defined by the equation:

x2 y2 z2 1

This is a centered sphere with its center at the origin (0, 0, 0) and a radius of 1 unit.

Plane

The plane is defined by the equation:

x y z 1

Our goal is to find the equation of the circle formed at the intersection of these two geometric entities. To achieve this, we will first identify the foot of the perpendicular from the center of the sphere to the given plane, then use this information to derive the equation of the circle.

Finding the Foot of the Perpendicular

Let M be the foot of the perpendicular from the center O (0, 0, 0) to the plane x y z 1. The position vector of M can be written as (x, y, z), and we need to determine its coordinates.

Calculating the Distance OM

The distance from the center O to the plane is given by the formula:

OM p |a0 a1y0 a2z0 d| / √(a02 a12 a22)

In our scenario, the plane equation is x y z - 1 0, so:

OM |1| / √(12 12 12) 1 / √3 1/√3

Equation of the Line OM

The line OM can be parameterized by the line equation:

x/1 y/1 z/1 t

Thus, any point M on this line can be represented as:

(x, y, z) (t, t, t)

Finding the Foot of the Perpendicular M

Since M lies on the plane x y z 1, substituting the parametric coordinates into the plane equation:

t t t 1

Solving for t, we get:

3t 1 > t 1/3

Therefore, the coordinates of point M (the foot of the perpendicular) are:

(1/3, 1/3, 1/3)

This point serves as the center of the circle formed by the intersection of the sphere and the plane.

Calculating the Radius of the Circle

The radius of the circle formed by the intersection can be found using the Pythagorean theorem in the right triangle formed by the radius of the sphere, the distance from the center of the sphere to the plane (OM), and the radius of the intersection circle.

Radius of the Intersection Circle (r')

Using the right triangle relationship:

r' √(r2 - p2)

Substituting the known values:

r' √(12 - (1/√3)2) √(1 - 1/3) √(2/3) √(2) / √3 √(2) / √3

Thus, the radius of the circle formed by the intersection is:

r' √(2) / √3

Conclusion

In summary, the equation of the circle formed by the intersection of the sphere x2 y2 z2 1 and the plane x y z 1 has a center at (1/3, 1/3, 1/3) and a radius of √(2) / √3.

Key Takeaways:

Understanding Geometric Relationships: The intersection of a sphere and a plane can result in a circle, which is a broader topic in geometry. Algebraic and Geometric Methods: Using algebraic equations to describe geometric entities is a fundamental technique in mathematics. Applications: Such concepts find applications in various fields, including computer graphics, physics, and engineering.

Frequently Asked Questions (FAQs)

Q1: What is the significance of the equation x y z 1 in the context of a sphere?

A1: The equation x y z 1 represents a plane that intersects the sphere x2 y2 z2 1, forming a circle. The plane can be positioned at any orientation relative to the sphere, and the equation describes the relationship between the plane's position and the sphere.

Q2: How can we use this knowledge in real-life applications?

A2: This knowledge can be applied in various fields. For instance, in computer graphics, understanding such intersections helps in rendering scenes with spheres and planes. In physics, it can be used to model interactions between spherical objects and planar surfaces.

Q3: Can this method be generalized to other geometric shapes?

A3: Yes, similar methods can be used to find the intersection of other geometric shapes. For example, finding the intersection of a cylinder and a plane involves similar steps, though the equations and methods differ slightly.