Calculator Inputs
Enter numeric coordinates, a symbolic spherical function, or both. The calculator keeps the chosen angle convention visible.
Formula Used
For the common physics convention, rho is distance from the origin. Theta is the azimuth angle in the xy-plane. Phi is measured down from the positive z-axis.
For the elevation convention, phi is measured upward from the xy-plane.
For symbolic functions, the calculator replaces spherical variables with Cartesian expressions. It uses rho = sqrt(x^2 + y^2 + z^2) and theta = atan2(y, x). The phi replacement depends on your selected convention.
How to Use This Calculator
Choose the conversion mode first. Select point mode for numeric coordinate conversion. Select function mode when you need an equation rewrite. Pick the correct phi convention before entering values. Enter theta and phi in the selected angle unit. Type the right side of the spherical function with names like rho, theta, and phi. Press the convert button. The result appears above the form, directly below the header area.
Understanding Spherical Function Conversion
Spherical functions describe points by distance and two angles. That view is useful for balls, cones, waves, fields, and surfaces around an origin. Cartesian functions use x, y, and z instead. They are often easier to graph on rectangular axes. They also match many algebra systems.
Why the convention matters
The same symbol can mean different angles. In many physics books, phi starts at the positive z-axis. In some math tools, phi is elevation from the xy-plane. A correct conversion must know this choice. Otherwise z may be swapped with the radial part in the xy-plane.
Point conversion
A numeric conversion starts with rho, theta, and phi. The calculator changes angles to radians internally when needed. Then it applies the selected formulas. The result gives x, y, and z. Decimal control helps with homework, reports, and engineering notes.
Function conversion
A spherical function may look like rho equals an expression in theta and phi. To rewrite it, replace rho with the Cartesian distance. Replace theta with the azimuth expression atan2(y, x). Replace phi with the matching polar or elevation expression. This produces a Cartesian relation.
Use with surfaces
Many surfaces have compact spherical forms. A sphere may use a constant rho. A cone may use a constant phi. A vertical half-plane may use a constant theta. After conversion, these shapes become equations in x, y, and z. Some forms can be simplified further by squaring or using trig identities.
Domain and accuracy
Spherical coordinates are not unique everywhere. Theta is not fixed on the z-axis. Phi may also be limited by convention. Most systems use a nonnegative radius. When a formula contains atan2 or arccos, domain checks matter. Rounding affects only displayed numeric answers. Symbolic expressions stay as typed replacements.
Practical workflow
Start with the convention used by your textbook or software. Convert one test point first. Then convert the full function. Compare the result with known geometry. If the equation looks complex, simplify common trig pairs. Keep restrictions with the final answer. This prevents valid points from being lost.
Common input checks
Use numeric values for point mode. Leave no angle field empty. Pick a distance label that matches your problem. Negative radius values can describe the same physical point with shifted angles. Many courses avoid that choice. The warning box highlights this case. It also notes when the point is near the vertical axis.
Reading symbolic output
The symbolic line is a relation, not always a solved function. It may contain inverse trig terms. This is normal for direct substitution. You can simplify after checking domains. For example, constant radius often becomes a sphere. Constant theta often becomes a plane through the z-axis. Constant phi often becomes a cone. Square carefully, because squaring may add extra branches. Compare the final relation with the original geometry. Check it before class work or reports too.
Frequently Asked Questions
What does this calculator convert?
It converts spherical points and spherical function expressions into Cartesian form. It can show numeric coordinates, symbolic substitutions, formulas, and domain notes.
Which spherical convention should I choose?
Choose the convention used by your class, book, or software. Physics convention measures phi from the positive z-axis. Elevation convention measures phi from the xy-plane.
Can I enter degrees?
Yes. Select degrees in the angle unit field. The calculator changes degrees to radians internally before calculating the Cartesian point.
Can I enter radians?
Yes. Select radians when your theta and phi values are already in radians. The calculator will use them directly for numeric conversion.
Does it simplify every symbolic equation?
No. It performs clear variable substitution. Some expressions may need manual algebra, squaring, factoring, or trigonometric simplification after conversion.
What is rho in Cartesian form?
Rho is the distance from the origin. In Cartesian variables, it becomes sqrt(x^2 + y^2 + z^2).
What is theta in Cartesian form?
Theta is the azimuth angle around the z-axis. The calculator represents it as atan2(y, x), which preserves quadrant information.
Why is theta sometimes undefined?
Theta is undefined on the z-axis because x and y are both zero there. Any azimuth angle points to the same vertical line.
Can this convert a constant rho sphere?
Yes. Enter rho on the left side and a constant on the right side. The result shows the Cartesian distance equal to that constant.
Can this convert cone equations?
Yes. A cone often has a constant phi. Choose the correct convention first, then convert phi equals that constant or related expression.
Why keep domain notes?
Domain notes protect the meaning of inverse trig functions and nonunique coordinates. They also remind users about radius and axis restrictions.