Surface Calculator
Convert a rectangular point to spherical values, or generate a valid point from spherical angles.
Example Data Table
| Point (x, y, z) | ρ | θ | φ | y z | 4√(x² + y²) | Result |
|---|---|---|---|---|---|---|
| (0, 4, 4) | √32 | 90° | 45° | 16 | 16 | Fits |
| (3, 4, 5) | √50 | 53.13° | 45° | 20 | 20 | Fits |
| (0, -4, -4) | √32 | 270° | 135° | 16 | 16 | Fits |
| (0, 4, 1) | √17 | 90° | 75.96° | 4 | 16 | Does not fit |
Formula Used
Use the standard spherical-coordinate identities below. Here, θ is the angle around the z axis, and φ is measured from the positive z axis.
ρ = 4 csc(θ) sec(φ)
csc(θ) = √(x2 + y2) / y and sec(φ) = ρ / z
ρ = 4 [√(x2 + y2) / y] [ρ / z]
y z = 4√(x2 + y2)
The squared alternative is y2z2 = 16(x2 + y2). Keep y z positive, because squaring can create extra points.
How to Use This Calculator
- Select Rectangular point to spherical values when x, y, and z are known.
- Enter the three coordinates. Use the sample button for the point (0, 4, 4).
- Choose a precision and select Calculate Surface.
- Read ρ, θ, φ, both rectangular sides, and the residual above the form.
- Use Spherical angles to surface point when θ and φ are known instead.
- Download the data as CSV or print the result for a PDF copy.
Understanding the Converted Surface
The relation rho equals four cosecant theta secant phi describes a spherical-coordinate surface. It becomes easier to inspect after conversion into rectangular coordinates. Use the standard identities for the horizontal radius, sine of theta, and cosine of phi. The horizontal radius is the square root of x squared plus y squared. Sine theta is y divided by that horizontal radius. Cosine phi is z divided by rho. Substitution removes the spherical variables. The result is y times z equals four times the horizontal radius. This form directly connects the Cartesian coordinates. It also reveals that the surface needs matching signs for y and z. A point with positive y needs positive z. A point with negative y needs negative z. Points on either coordinate plane are not allowed because the original reciprocal functions are undefined there.
Geometry and Domain Restrictions
Squaring the converted equation gives y squared z squared equals sixteen times x squared plus y squared. This squared form may look simpler for graphing software. However, it can include extra points. Squaring removes the sign information from the original relation. The unsquared equation is therefore the safer primary condition. Keep y z positive, and keep both y and z nonzero. The calculator reports these domain conditions automatically. It also computes rho, theta, and phi from an entered point. Theta is measured around the z axis. Phi is measured downward from the positive z axis. These conventions are common in calculus and engineering. Different books sometimes exchange the angle names. Always check the convention before comparing a result.
Checking Real Coordinate Points
To test a rectangular point, enter x, y, and z. The calculator finds the radial distance rho. It then calculates theta with the two argument arctangent. This prevents quadrant errors. Next, it finds phi from the vertical coordinate. The tool compares y z with four times the horizontal radius. A zero residual means the point fits the converted surface, subject to rounding. The sample point zero, four, four is useful. Its product y z is sixteen. Its horizontal radius is four. Four times that radius is also sixteen. The spherical values are rho equals square root of thirty two, theta equals ninety degrees, and phi equals forty five degrees. The original spherical expression returns the same radial distance. This confirms both descriptions represent the same point.
Graphing and Practical Use
The surface is useful when studying transformations between coordinate systems. Near y equals zero or z equals zero, the reciprocal factors become very large. Those regions are excluded, not merely inconvenient. Small residuals can reflect normal rounding when necessary. Large residuals clearly show that the point is away from the surface. Choose the spherical input mode to generate a point from theta and phi. The calculator rejects undefined angles and nonpositive standard radii. Export the displayed values as CSV for records. Print the result page when a PDF copy is needed. Use calculated results to graph this curved surface correctly.
Frequently Asked Questions
1. What is the primary rectangular equation?
The primary equivalent is y z = 4√(x2 + y2). It preserves the sign condition needed by the original reciprocal functions.
2. Why are y and z not allowed to be zero?
When y is zero, sin(θ) is zero and csc(θ) is undefined. When z is zero, cos(φ) is zero and sec(φ) is undefined.
3. Is the squared equation fully equivalent?
No. The squared form can admit points with y z less than zero. Keep the unsquared equation and the positive product condition for the original relation.
4. Which spherical angle convention does this page use?
Theta rotates around the z axis. Phi starts on the positive z axis and moves toward the negative z axis.
5. What does the residual measure?
The residual is y z minus 4√(x2 + y2). A value near zero shows that the tested point is on the converted surface.
6. Can the coordinates be negative?
Yes. Valid points may use negative y and negative z together. Their product must remain positive, and neither coordinate may be zero.
7. Why can a spherical angle pair be rejected?
The expression needs nonzero sin(θ) and cos(φ). Standard nonnegative rho also needs those two quantities to have matching signs.
8. Can I enter angles outside 0 to 360 degrees?
Yes for theta. The calculator normalizes theta to one full rotation. Phi must stay strictly between 0 and 180 degrees.
9. Is x restricted by the equation?
No direct x limit appears. Its value affects the horizontal radius, which then sets the required relationship between y and z.
10. Can I use this result in graphing software?
Yes. Enter y z = 4√(x2 + y2) and preserve the restrictions y z greater than zero, y not zero, and z not zero.
11. Which sample point gives a quick check?
The point (0, 4, 4) works because y z and four horizontal radii both equal sixteen. Use calculated results to graph this curved surface correctly.