Local Flatness of a Sphere in 3-D Euclidean Space
In 3-D Euclidean space, a sphere can be locally approximated as a plane. The proof is as follows.
Notation
| Symbol | Type | Explanation |
|---|---|---|
| \(R\) | \(\mathbb{R}^+\) | radius of the sphere |
| \(x, y, z\) | \(\mathbb{R}\) | coordinates on the sphere |
| \(P(x_0, y_0, z_0)\) | \(\mathbb{R}^3\) | point on the sphere's surface |
| \(C(a, b, c)\) | \(\mathbb{R}^3\) | center of the sphere |
| \(x_0, y_0, z_0\) | \(\mathbb{R}\) | coordinates of \(P(x_0, y_0, z_0)\) |
| \(a, b, c\) | \(\mathbb{R}\) | coordinates of \(C(a, b, c)\) |
| \(\mathbf{n}\) | \(\mathbb{R}^3\) | normal vector at \(P\) |
| \(A, B, C\) | \(\mathbb{R}\) | components of \(\mathbf{n}\) |
| \(\Delta x, \Delta y, \Delta z\) | \(\mathbb{R}\) | small perturbations at \(P\) |
| \(\approx\) | relation | approximation |
| \((\Delta x)^2, (\Delta y)^2, (\Delta z)^2\) | \(\mathbb{R}\) | second-order infinitesimals |
Proof
1. Sphere in 3-D Euclidean Space
Consider a sphere with center \(C(a, b, c)\) and radius \(R\) in 3-dimensional Euclidean space. The equation of the sphere is: \[ (x-a)^2+(y-b)^2+(z-c)^2=R^2 \] where \((x, y, z)\) are the coordinates of any point on the surface of the sphere.
2. Tangent Plane to the Sphere
Let \(P\left(x_0, y_0, z_0\right)\) be a point on the sphere's surface. The tangent plane at \(P\) is the plane that touches the sphere at exactly one point, \(P\), and is perpendicular to the radius of the sphere at \(P\).
To find the equation of the tangent plane at \(P\left(x_0, y_0, z_0\right)\), observe that the vector from the center of the sphere \(C(a, b, c)\) to the point \(P\left(x_0, y_0, z_0\right)\) is given by: \[ \mathbf{n}=\left(x_0-a, y_0-b, z_0-c\right) \]
This vector is the normal (perpendicular) vector to the tangent plane at \(P\). The general equation of a plane with normal vector \(\mathbf{n}=(A, B, C)\) passing through a point \(\left(x_0, y_0, z_0\right)\) is: \[ A\left(x-x_0\right)+B\left(y-y_0\right)+C\left(z-z_0\right)=0 \]
Substituting the components of the normal vector \(\mathbf{n}\) into the equation, the equation of the tangent plane at \(P\left(x_0, y_0, z_0\right)\) becomes: \[ \left(x_0-a\right)\left(x-x_0\right)+\left(y_0-b\right)\left(y-y_0\right)+\left(z_0-c\right)\left(z-z_0\right)=0 \]
3. Simplifying the Tangent Plane Equation
Expanding and rearranging the equation, we get: \[ \left(x_0-a\right) x+\left(y_0-b\right) y+\left(z_0-c\right) z=\left(x_0-a\right) x_0+\left(y_0-b\right) y_0+\left(z_0-c\right) z_0 \]
This is the equation of the tangent plane to the sphere at \(P\left(x_0, y_0, z_0\right)\).
4. Local Approximation
Consider any point \(Q(x, y, z)\) on the sphere around the point \(P\left(x_0, y_0, z_0\right)\), where \[ x=x_0+\Delta x, \quad y=y_0+\Delta y, \quad z=z_0+\Delta z . \]
Since \(Q(x, y, z)\) is on the sphere, we get \[ \left(x_0+\Delta x-a\right)^2+\left(y_0+\Delta y-b\right)^2+\left(z_0+\Delta z-c\right)^2=R^2, \] which we further expand to \[ (x_0 - a)^2 + 2(x_0 - a)\Delta x + (\Delta x)^2 + (y_0 - b)^2 + 2(y_0 - b)\Delta y + (\Delta y)^2 + (z_0 - c)^2 + 2(z_0 - c)\Delta z + (\Delta z)^2 = R^2 . \] Notice that the terms \[ (x_0 - a)^2 + (y_0 - b)^2 + (z_0 - c)^2 = R^2 \] because \(P(x_0, y_0, z_0)\) lies on the sphere, so these terms cancel out with the \(R^2\) on the right-hand side. We are left with: \[ 2(x_0 - a)\Delta x + 2(y_0 - b)\Delta y + 2(z_0 - c)\Delta z + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 = 0 \] The terms \((\Delta x)^2,(\Delta y)^2\), and \((\Delta z)^2\) are second-order infinitesimals, meaning they become negligible when \(\Delta x, \Delta y\), and \(\Delta z\) are very small.
Therefore, we can approximate the equation by neglecting these second-order terms: \[ 2\left(x_0-a\right) \Delta x+2\left(y_0-b\right) \Delta y+2\left(z_0-c\right) \Delta z \approx 0 \] Dividing the above equation by 2 gives the linear approximation: \[ \left(x_0-a\right) \Delta x+\left(y_0-b\right) \Delta y+\left(z_0-c\right) \Delta z=0 \]
Replace \(x\), \(y\), \(z\) with \(x_0+\Delta x\), \(y_0+\Delta y\), \(z_0+\Delta z\): \[ \left(x_0-a\right)\left(x-x_0\right)+\left(y_0-b\right)\left(y-y_0\right)+\left(z_0-c\right)\left(z-z_0\right)=0 \] This equation is precisely the tangent plane equation through the point \(P\left(x_0, y_0, z_0\right)\).
Therefore, near any point \(P\left(x_0, y_0, z_0\right)\) on the sphere, the sphere's equation can be approximated by the tangent plane equation through \(P\left(x_0, y_0, z_0\right)\).