Sphere Graphing Calculator

Sphere Equation:

\[ (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2 \]

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1. What is a Sphere in 3D Space?

A sphere is a perfectly symmetrical three-dimensional shape where every point on its surface is equidistant from its center. In mathematics, it's defined as the set of all points in 3D space that are at a given distance (radius) from a fixed point (center).

2. How Does the Calculator Work?

The calculator generates a 3D plot of a sphere using parametric equations:

\[ \begin{cases} x = x_0 + r \cdot \sin\phi \cdot \cos\theta \\ y = y_0 + r \cdot \sin\phi \cdot \sin\theta \\ z = z_0 + r \cdot \cos\phi \end{cases} \]

Where:

\( (x_0, y_0, z_0) \) — Center coordinates
\( r \) — Radius of the sphere
\( \theta \) — Azimuthal angle (0 to 2π)
\( \phi \) — Polar angle (0 to π)

3. Understanding the Sphere Equation

The standard equation of a sphere in 3D Cartesian coordinates is:

\[ (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2 \]

This equation states that for any point (x, y, z) on the sphere's surface, the squared distance to the center equals the squared radius.

4. Using the Calculator

Instructions: Enter the center coordinates (x₀, y₀, z₀) and radius of the sphere. The calculator will generate an interactive 3D plot that you can rotate and zoom.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between a sphere and a circle?
A: A circle is a 2D shape, while a sphere is its 3D counterpart. A sphere has surface area and volume, while a circle has circumference and area.

Q2: How is this different from a ball?
A: In mathematics, a sphere refers to the surface only, while a ball includes both the surface and all points inside.

Q3: Can I plot multiple spheres?
A: This calculator plots one sphere at a time. For multiple spheres, you would need to combine the data points.

Q4: Why does my sphere look pixelated?
A: The resolution depends on the number of points calculated. More points create smoother spheres but require more computation.

Q5: How can I save the plot?
A: Use the camera icon in the plot toolbar to download the image as PNG or export the data.