Cartesian to Spherical Coordinates

The convention used by this converter

Several spherical-coordinate conventions exist, especially for which angle is called polar or azimuth. This page fixes the convention before calculating. Rho is the distance from the Cartesian origin. Azimuth is the direction of the point's projection onto the xy plane, measured counterclockwise from positive x. Polar angle is measured from positive z toward the point, so it is zero on the positive z-axis, 90 degrees in the xy plane, and 180 degrees on the negative z-axis. The result labels repeat those meanings.

The convention is useful because it separates radial scale from orientation. Two points on the same sphere have the same rho but can have different angles. Two points on the same ray have the same azimuth and polar angle but can have different rho. The converter does not infer a physical latitude, longitude, compass bearing, or application-specific orientation. If another system uses a different axis or angle convention, transform the result deliberately and document that change rather than renaming an output silently.

• rho measures distance from the origin.
• Azimuth starts at positive x in the xy plane.
• Polar angle starts at positive z.
• The page uses degrees for both angular outputs.

Read the Cartesian inputs

Enter the three coordinates x, y, and z in one shared Cartesian frame. The coordinate values may be positive, negative, or fractional within the displayed range. Their length unit is carried to rho, while the angles are unitless quantities displayed in degrees. The fields are not radius and angles in disguise; they are the original rectangular components from which the radial and directional quantities are derived.

Each coordinate must be a finite number between -1,000,000 and 1,000,000 inclusive. These limits are numerical safety boundaries for a browser calculator and do not define the extent of Cartesian space. The direct handler applies the same rules as the form, rejecting missing values, numeric strings, NaN, infinities, and out-of-range entries. A coordinate of zero is valid. Only the simultaneous triple (0, 0, 0) is excluded because it has no direction.

• The inputs are Cartesian x, y, and z components.
• Signed and decimal values are valid within bounds.
• rho keeps the coordinate length unit.
• The origin is the one model-specific singular point.

Deriving radial distance

The radial distance is the ordinary Euclidean distance from the origin to the point: rho = sqrt(x^2 + y^2 + z^2). It is the hypotenuse of the three-dimensional right-triangle construction. First project the point onto the xy plane; the projection has length sqrt(x^2 + y^2). Then combine that horizontal projection with z using another Pythagorean step. The resulting rho is nonnegative and has the same length unit as each Cartesian coordinate.

The radial value remains defined for every nonzero input triple, including points on an axis and points with negative coordinates. Squaring removes direction only for the purpose of measuring distance. It does not mean that the original point has lost its quadrant; azimuth and polar angle preserve orientation. The engine uses a finite hypotenuse calculation and checks the result before returning it, while the zero check occurs before dividing z by rho for the polar angle.

• rho = sqrt(x^2 + y^2 + z^2).
• Radial distance is always nonnegative.
• Negative coordinates affect angles but not squared radial contributions.
• rho must be nonzero before angular division.

Azimuth and quadrant handling

Azimuth is computed with atan2(y, x), not with a one-argument arctangent. The two-argument function uses the signs of both x and y to identify the correct quadrant. A point with x positive and y positive has an azimuth between 0 and 90 degrees. Negative y produces a clockwise-looking raw negative angle, which the handler adds to 360 so the displayed range is nonnegative. Points on the negative x-axis receive 180 degrees, and points on the positive x-axis receive zero.

At x = 0 and y = 0, the xy projection has no direction, so azimuth is mathematically undefined even when z is nonzero. The current converter returns the normalized result supplied by atan2 for that axis case, while the overall point still has a defined polar direction. The origin is the only input rejected at the model level because both angles are undefined there. Downstream applications that require a unique azimuth at the z-axis should add a convention such as zero and state it explicitly.

• Use atan2(y, x) to preserve the quadrant.
• Azimuth is normalized to 0 through 360 degrees.
• Negative raw angles are represented by their positive equivalent.
• A downstream z-axis convention should be stated if required.

Polar angle from the positive z-axis

The polar angle is theta = acos(z/rho). Since rho is the point's full radial distance, z/rho is the cosine of the angle between the positive z-axis and the point vector. A point above the xy plane has a polar angle below 90 degrees; a point below it has an angle above 90 degrees. The positive z-axis is zero degrees, any nonzero point in the xy plane is 90 degrees, and the negative z-axis is 180 degrees.

Finite floating-point arithmetic can make z/rho differ from the mathematical interval [-1, 1] by a tiny amount at an axis endpoint. The handler clamps the ratio to that interval before acos, while the input and radial result remain checked. This is endpoint hygiene, not a correction of materially invalid data. The output is in degrees so it can be read with the azimuth result; the calculation steps retain the cosine ratio that explains how the angle was obtained.

• Polar angle uses acos(z/rho).
• It is measured from positive z, not from the xy plane.
• The valid output range is 0 through 180 degrees.
• Tiny endpoint round-off is clamped before acos.

Worked example at equal coordinates

For x = 1, y = 1, and z = 1, rho = sqrt(1 + 1 + 1) = sqrt(3), approximately 1.73205 length units. The xy projection lies equally along positive x and positive y, so atan2(1, 1) gives azimuth 45 degrees. The polar cosine is 1/sqrt(3), and acos of that ratio gives approximately 54.7356 degrees. The point is above the xy plane and halfway between the positive x and y directions in its horizontal projection, which agrees with both angular outputs.

A useful scale check is to multiply all three coordinates by two. Rho doubles, but azimuth and polar angle remain unchanged because the ray from the origin is unchanged. Reflecting y to -1 changes azimuth to 315 degrees while leaving rho and polar angle unchanged. Reflecting z to -1 preserves rho and azimuth but changes polar angle to its supplement. These transformations make the separate roles of scale, horizontal direction, and vertical direction visible.

• (1, 1, 1) has rho = sqrt(3).
• Its azimuth is 45 degrees.
• Its polar angle is about 54.7356 degrees.
• Uniform scaling changes rho but not the angles.

Axes, signs, and singular behavior

Axis points provide exact convention checks. A point on positive x has azimuth 0 and polar angle 90 degrees. A point on negative x has azimuth 180 and polar angle 90. A point on positive y has azimuth 90, and negative y is normalized to 270. Positive and negative z points have radial distance equal to the absolute z value, an axis azimuth convention, and polar angle 0 or 180. These cases should be tested when integrating the converter into graphics or a physics calculation.

The origin is different from an axis point. At the origin, rho is zero and the ratios used for angles have no defined direction. The handler rejects it with an explicit error rather than returning arbitrary zeros. A nonzero point on the z-axis has a perfectly defined ray and radial distance, even though its xy projection has zero length. Whether an application assigns azimuth zero on that ray is a convention outside the core formula and should not be confused with the origin singularity.

• Positive x is the azimuth reference direction.
• Negative y is displayed as 270 degrees.
• Positive and negative z distinguish polar endpoints.
• Only the zero vector is rejected by the core converter.

Units, bounds, and finite output

The three coordinates share a length unit, and rho uses that same unit. Azimuth and polar angle are degrees regardless of the length scale. The page performs no conversion between metres, inches, pixels, or other units, and it does not infer whether the frame is right-handed from the values. A left-handed graphics convention may reverse an orientation interpretation even though the arithmetic returns numbers. Normalize the source frame before entry when a downstream model has a strict convention.

The coordinate bounds are inclusive at plus and minus 1,000,000. The handler validates type, finiteness, and range, then checks rho, atan2, the cosine ratio, and polar angle. Results are labeled and returned as finite numbers with no negative zero. The renderer may round displayed angles or rho, but a later transformation should use the underlying values. A very small nonzero point can have well-defined angles while its displayed rho rounds to zero, so keep the distinction between display and numeric state.

• rho preserves the shared Cartesian length unit.
• Angles are explicitly reported in degrees.
• All coordinate and derived values receive finite checks.
• Display rounding does not change the numeric conversion.

Related coordinates and non-equivalent outputs

Spherical coordinates are not latitude and longitude on a geographic body unless an application defines a radius and an axis convention that makes them so. Polar angle and geographic latitude are complementary under some conventions, while azimuth and longitude may use different zero directions or signs. A great-circle calculator operates on angular surface positions and a sphere radius; this converter operates on a Cartesian point and returns its ray representation. Similar angle names do not make the fields interchangeable.

A cylindrical-coordinate conversion would return a distance in the xy plane, an azimuth, and z unchanged. It answers a different question and has a different singular axis behavior. Cartesian-to-spherical conversion also does not compute a vector's velocity, field direction, or physical orientation unless the entered components represent that quantity. Keep the coordinate transformation separate from any later law or simulation that consumes the three outputs.

• Geographic latitude is not automatically polar angle.
• Great-circle inputs and Cartesian inputs are different contracts.
• Cylindrical coordinates use a different radial quantity.
• The conversion does not infer physical meaning from component names.

Reproducible handoff and safe use

A reproducible conversion record includes x, y, z, the handedness and axis convention, the length unit, and the definitions of both angles. Recompute rho, atan2, and acos(z/rho) independently when reviewing an integration. Test an equal-coordinate point, a negative-y point, a positive-z point, and the origin rejection. If the result is passed to a renderer or simulation, preserve whether angles are degrees or radians; this page deliberately returns degrees even though its trigonometric internals use radians.

The calculator is useful for analytic geometry, graphics coordinate checks, spherical parameter examples, and education. It does not validate a sensor frame, a geodetic datum, a transformation matrix, or a physical unit calibration. In robotics, navigation, imaging, or engineering, confirm the frame and uncertainty with the system that owns the measurements. Enter only the three required numeric components and keep sensitive spatial context out of a shared record when the point could identify a person, site, or asset.

• Record frame, handedness, units, and angle definitions.
• Test quadrant, axis, scaling, and origin behavior.
• Convert degrees and radians explicitly at system boundaries.
• Use domain validation for measured or safety-critical coordinates.

Checking a conversion by reconstructing components

The spherical values can be checked by reconstructing the original Cartesian direction. With rho, azimuth alpha, and polar angle theta, the usual convention gives x = rho sin(theta) cos(alpha), y = rho sin(theta) sin(alpha), and z = rho cos(theta), after converting the displayed degree angles back to radians. The converter does not perform this inverse step, but it is a useful independent review. Rounding the angles before reconstruction can create a small residual even when the forward conversion is correct.

A second check uses invariants. Rho should equal the Euclidean distance from the origin, changing sign of every Cartesian coordinate should preserve rho, and uniform positive scaling should preserve both angles. Reflecting across the xy plane should replace polar angle theta with 180 - theta, while reflecting y changes azimuth's sign convention. These relationships test the convention more thoroughly than checking one decimal output.

• Inverse formulas can reconstruct x, y, and z.
• Rho equals the Cartesian Euclidean norm.
• Uniform positive scaling preserves both angles.
• Angle rounding can create reconstruction residuals.

Frames, handedness, and downstream interfaces

A coordinate conversion is only meaningful inside a declared frame. Graphics packages may place positive z toward or away from the viewer, while scientific systems may use a right-handed or left-handed basis. The same three numeric components can therefore receive different directional interpretations even though rho is unchanged. This page fixes the algebraic convention but does not inspect a matrix or infer handedness. Apply any frame transformation before conversion when the source and destination conventions differ.

The angular outputs are degrees by design. Many libraries and simulation APIs expect radians, so convert exactly once at the boundary and label the converted values. A downstream system should also state whether it wants azimuth in a signed interval such as -180 through 180 or the normalized 0 through 360 interval returned here. Changing that representation is a presentation or interface operation, not a change to the point's radial geometry.

• Declare handedness and positive-axis directions.
• Transform frames before applying this convention.
• Convert degrees to radians at explicit interfaces.
• Choose signed or normalized azimuth deliberately downstream.

Reviewing singular and near-axis inputs

Near the origin, rho is small and a tiny change in x, y, or z can cause a large change in the reported angles. That sensitivity is geometric, not necessarily a software defect. The origin itself has no ray and is rejected. Near the positive or negative z-axis, the azimuth is also poorly determined because the xy projection is tiny; an application that uses azimuth for control should define a tolerance and a stable axis convention rather than relying on many displayed decimal places.

Near an axis, inspect the raw Cartesian values and rho alongside the angles. A rounded azimuth of zero does not prove that y was exactly zero, and a rounded polar angle of 90 does not prove that z was exactly zero. Preserve the numeric input and the displayed precision separately. The calculator reports the deterministic mathematical conversion; uncertainty and conditioning analysis belong to the workflow that supplied the point.

• Angles are sensitive near the origin and z-axis.
• The exact origin is rejected as undefined.
• Rounded axis angles do not prove exact axis inputs.
• Conditioning and uncertainty need a downstream policy.

Round-trip checks at a system boundary

A coordinate pipeline can test the conversion by sending a non-origin Cartesian point through this page, converting the degree angles to radians, and reconstructing the components with the inverse spherical equations. The reconstructed values should agree within the tolerance caused by numeric precision and any intentional display rounding. Use the original unrounded rho and angles for this check. A mismatch concentrated in x and y often signals an azimuth convention error, while a mismatch in z often signals a polar-angle reference error.

The point's norm is an especially useful invariant. Reconstructing x, y, and z should give x squared plus y squared plus z squared close to rho squared. A uniform scale factor should change only rho. Rotating the point around the z-axis should change azimuth while preserving rho and polar angle. These checks exercise the semantics of all three outputs and help distinguish a bad conversion from a bad downstream frame transform.

At the positive x-axis, the expected outputs are rho equal to the positive x magnitude, azimuth zero, and polar angle 90 degrees. At the positive z-axis, rho equals z, polar angle is zero, and the azimuth follows the axis convention supplied by atan2. Use these axis cases deliberately rather than treating their angle representation as a universal physical truth. A consuming application may need to handle the axis as a special equivalence class.

Do not compare a spherical output with geographic coordinates without defining the relationship between the Cartesian frame and the body's surface. A Cartesian point at radius rho may be inside, outside, or on a sphere chosen by another system. The converter reports geometry about its origin; it does not classify a location, compute a surface distance, or apply a map datum.

For a reproducible interface, transmit named values and units: rho in the common length unit, azimuth in normalized degrees, and polar angle in degrees from positive z. If the next system uses radians or a signed azimuth, make that transformation explicit in its adapter. This preserves the converter's contract while preventing a correct numeric result from being consumed under a different angle convention.

The most important review question is not whether three numbers were produced, but whether the consuming system uses the same positive axes and angle origins. Preserve that convention beside every converted point. A mathematically valid conversion with an unannounced axis reversal can look plausible while pointing in the wrong direction.

• Use inverse equations for a round-trip check.
• Verify norm, scale, rotation, and axis invariants.
• Treat axis-angle equivalence deliberately.
• Do not equate Cartesian spherical output with geographic location.
• Adapt units and angle ranges explicitly at interfaces.