Goal
Convert Cartesian x, y, z coordinates to cylindrical rho, normalized azimuth, and unchanged z.
Cylindrical Coordinates
Convert Cartesian x, y, z coordinates to cylindrical rho, normalized azimuth, and unchanged z.
Key facts
- What it does
- Convert Cartesian x, y, z coordinates to cylindrical rho, normalized azimuth, and unchanged z.
- Formula
rho=sqrt(x^2+y^2), azimuth=atan2(y,x) normalized to [0,360) degrees, and z_cylindrical=z; azimuth is undefined when x=y=0.- You enter
- Cartesian x · Cartesian y · Cartesian z
- Worked example
- rho=sqrt(1^2+1^2)=sqrt(2), azimuth=45 degrees, and cylindrical z remains 3 coordinate units.
A clearer path to an answer
From your question to a useful result
This page keeps the calculation transparent: define the goal, enter the matching values, inspect the method, and decide what the result means in your situation.
Inputs
Cartesian x · Cartesian y · Cartesian z
Method
rho=sqrt(x^2+y^2), azimuth=atan2(y,x) normalized to [0,360) degrees, and z_cylindrical=z; azimuth is undefined when x=y=0.
Next step
Calculate, review the assumptions below, then compare a related tool when the decision needs more context.
Cylindrical Coordinates
Convert Cartesian x, y, z coordinates to cylindrical rho, normalized azimuth, and unchanged z.
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Result
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Calculation map
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01
Inputs (3)
- Cartesian x Ready
- Cartesian y Ready
- Cartesian z Ready
02
Formula
rho=sqrt(x^2+y^2), azimuth=atan2(y,x) normalized to [0,360) degrees, and z_cylindrical=z; azimuth is undefined when x=y=0.
Bounded, transparent calculation
03
Result
- Calculate to preview the result.
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Formula, assumptions, and example
Formula: rho=sqrt(x^2+y^2), azimuth=atan2(y,x) normalized to [0,360) degrees, and z_cylindrical=z; azimuth is undefined when x=y=0.
Cylindrical coordinates measure distance from the z-axis in the xy-plane, angle from the positive x-axis, and height along z. The azimuth is normalized to a half-open degree range and the z value is unchanged.
- The coordinates are finite points in one right-handed Cartesian system with compatible units.
- Azimuth is reported in degrees from 0 inclusive through 360 exclusive, and the positive z-axis is represented by rho=0 with no defined azimuth.
- This is a coordinate conversion only; it does not calculate spherical polar angle, velocity, or a physical orientation.
Worked example: rho=sqrt(1^2+1^2)=sqrt(2), azimuth=45 degrees, and cylindrical z remains 3 coordinate units.
Displayed input contract
- Cartesian x · minimum -1000000 · maximum 1000000
- Cartesian y · minimum -1000000 · maximum 1000000
- Cartesian z · minimum -1000000 · maximum 1000000
The displayed limits are checked before the handler runs. Model-specific domain checks may also reject impossible or non-finite inputs.
Methodology: This calculator follows the WorldCalculate input, formula, precision, and boundary policy. Read the official methodology.
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Answer-first guide
How to use the Cylindrical Coordinates for a real question
Convert Cartesian x, y, z coordinates to cylindrical rho, normalized azimuth, and unchanged z. Start with one clearly defined goal, enter values in the units shown, and keep the result attached to the assumptions below.
What this answers
This tool is useful when your question includes cylindrical coordinates, Cartesian conversion, rho. It returns the outputs declared in the calculator contract rather than a live quote, approval, diagnosis, or professional sign-off.
What you enter
Cartesian x · Cartesian y · Cartesian z. Keep the same time period, unit system, and currency wherever the form requires comparable values.
How to check it
Run the worked example first, compare its output with the page's example, then change one input at a time. This makes an unexpected result easier to trace to a unit, boundary, or assumption.
Three checks before you rely on the answer
- Match the question. Confirm that the result means the quantity you need, not a similar-sounding percentage, balance, rate, or estimate.
- Match the inputs. Use the requested units and period, and read each hint before replacing the example values with your own.
- Read the boundary. Review the assumptions and limits. The coordinates are finite points in one right-handed Cartesian system with compatible units.
Need a wider view? Browse Math Calculators or compare the related tools below. The WorldCalculate methodology explains how formulas, examples, limits, and revisions are reviewed.
How to use the Cylindrical Coordinates
- Enter Cartesian x — The Cartesian x-coordinate. (coordinate units).
- Enter Cartesian y — The Cartesian y-coordinate. (coordinate units).
- Enter Cartesian z — The Cartesian z-coordinate, preserved in the conversion. (coordinate units).
- Choose Calculate and read the result panel.
- Use Download PDF or Download Word to save a result sheet.
Formula
rho=sqrt(x^2+y^2), azimuth=atan2(y,x) normalized to [0,360) degrees, and z_cylindrical=z; azimuth is undefined when x=y=0.
Cylindrical coordinates measure distance from the z-axis in the xy-plane, angle from the positive x-axis, and height along z. The azimuth is normalized to a half-open degree range and the z value is unchanged.
Worked example
rho=sqrt(1^2+1^2)=sqrt(2), azimuth=45 degrees, and cylindrical z remains 3 coordinate units.
Assumptions and limits
- The coordinates are finite points in one right-handed Cartesian system with compatible units.
- Azimuth is reported in degrees from 0 inclusive through 360 exclusive, and the positive z-axis is represented by rho=0 with no defined azimuth.
- This is a coordinate conversion only; it does not calculate spherical polar angle, velocity, or a physical orientation.
Who uses this calculator?
- Students learning multiple coordinate systems
- Programmers converting planar radial data with an axial coordinate
- Physics and engineering learners checking cylindrical notation
When is it useful?
- Convert a Cartesian point to radial distance, azimuth, and height.
- Normalize negative atan2 angles for storage and display.
- Keep the singular z-axis case explicit instead of inventing an angle.
Context and background
The mathematical structure behind the tools
Math calculators move from named quantities to a relation, then to a result that can be checked with substitution, units, or an alternate form.
Arithmetic, algebra, geometry, trigonometry, and number theory provide reusable structures for classroom work and everyday reasoning. Each page narrows that structure to one declared problem.
Research and review
How this guide was researched
Researched by Hassan ALRowaie, Founder and editorial researcher at WorldCalculate.
This guide follows the live calculator's declared inputs, formula, worked example, assumptions, validation boundaries, and source-backed methodology. The review date describes editorial review of the calculator explanation; it is not a promise that external facts or rates remain current.
Cylindrical coordinates describe a three-dimensional Cartesian point by its distance from the z-axis, its direction around that axis, and its unchanged height. This calculator accepts x, y, and z, computes rho=sqrt(x^2+y^2), normalizes azimuth from atan2 to [0,360) degrees, and returns z unchanged. It rejects the z-axis case x=y=0 because an azimuth is not defined there. The guide distinguishes this conversion from spherical coordinates, explains quadrant and angle behavior, gives a worked example, records finite boundaries and assumptions, and sets a conservative limit against treating coordinates as a physical model without a declared frame.
The coordinate question answered
The page answers how to express one Cartesian point in cylindrical notation. A cylindrical triple is (rho, phi, z): rho is the nonnegative distance from the z-axis, phi is an angle around that axis, and z is the same axial coordinate used by the Cartesian input. The conversion changes the description, not the point. A later calculation can reconstruct x and y from rho and phi, but reconstruction is not performed by this handler.
The output angle is a convention, not a new measured length. This page measures phi from the positive x-axis and reports degrees in the half-open interval [0,360). That choice makes the representation deterministic for ordinary points, while the axis rho=0 remains singular for azimuth. The result does not select a different branch merely because a point lies in a particular quadrant.
- rho is distance from the z-axis.
- Azimuth is measured from positive x.
- z is preserved exactly as the axial coordinate.
- The conversion describes the same point in another coordinate system.
Read the Cartesian fields
Enter x and y as the coordinates in the plane perpendicular to the z-axis. Enter z as the axial coordinate. The fields accept negative values, zeros, and decimals within the displayed finite bounds. All three values must use one coordinate origin, axis orientation, and scale. If x and y come from one frame but z comes from another, the arithmetic will still accept the numbers while the resulting point meaning will be inconsistent.
The pure handler requires JavaScript numbers and repeats the form's finite bounds. Numeric strings, blanks, NaN, infinity, and out-of-range values are rejected. Zero is valid for any one coordinate. It is the combination x=0 and y=0 that creates the azimuth singularity, regardless of the z value. The axis can be represented as a point with nonzero height, but its angular coordinate remains undefined.
- x and y determine radial distance and azimuth.
- z passes through unchanged.
- Signed and fractional coordinates are allowed inside bounds.
- A nonzero z does not remove an x=y=0 azimuth singularity.
Computing rho in the xy-plane
The radial coordinate is rho=sqrt(x^2+y^2). It is the two-dimensional Euclidean distance from the origin of the xy-plane, which is the point where the z-axis pierces that plane. The z value is not included in this square root. This is the key difference from a spherical radial distance, which would also include z^2. A point can be far above the xy-plane while still having a small cylindrical rho if it is close to the z-axis.
The handler uses a stable hypotenuse operation for x and y and checks the resulting rho. The value is nonnegative by construction. If rho is zero, the handler rejects before calling the angular calculation because atan2(0,0) does not provide a unique direction. Otherwise, the radial distance is a finite length in the same unit as x and y.
- rho uses x and y only.
- Cylindrical rho is not the spherical distance from the origin.
- rho is always nonnegative.
- rho=0 triggers the undefined-azimuth rule.
Quadrants and azimuth normalization
atan2(y, x) uses the signs of both planar coordinates to identify the correct quadrant. A point with positive x and positive y has an angle between 0 and 90 degrees. Negative x with positive y belongs to the second quadrant, both negative belongs to the third, and positive x with negative y belongs to the fourth. The handler converts the radian result to degrees and adds 360 when the raw value is negative. An exact 360 is folded to zero so the output remains half-open.
Axis points provide useful checks. Positive x with y=0 yields azimuth zero. Positive y with x=0 yields 90. Negative x with y=0 yields 180, and negative y with x=0 yields 270. The angle is not normalized to -180 through 180 or to radians because the catalog contract says degrees from positive x. A downstream system using another convention must convert explicitly.
- atan2 preserves quadrant information.
- Negative raw angles are shifted by 360 degrees.
- The reported range is 0 <= azimuth < 360.
- Cardinal-axis values make good normalization checks.
A worked Cartesian example
Use x=1, y=1, and z=3. The planar radius is rho=sqrt(1^2+1^2)=sqrt(2), approximately 1.414213562. atan2(1,1) is pi/4 radians, which converts to 45 degrees. The cylindrical axial value is z=3. The result is therefore (sqrt(2),45,3), with the numeric rho and angle entries returned separately. The point is in the first quadrant and lies above the xy-plane because z is positive.
For a sign check, change x and y to -1 and -1 while keeping z=3. rho remains sqrt(2), but the normalized azimuth becomes 225 degrees. This shows that rho loses planar direction while azimuth retains it. Change only z to -8 and rho and azimuth stay the same while the axial result becomes -8. Those independent changes make the field roles visible without relying on an external diagram.
- (1,1,3) gives rho=sqrt(2).
- Its azimuth is 45 degrees.
- Its cylindrical z is 3.
- Changing planar signs changes azimuth but not rho.
The z-axis edge case
Every point with x=0 and y=0 lies on the z-axis. Its cylindrical rho is zero, and there is no preferred ray in the xy-plane from which to measure an azimuth. This remains true at z=0, at positive z, and at negative z. The handler rejects all of these inputs with an explicit undefined-azimuth error rather than returning zero, NaN, or an arbitrary angle. The rejection concerns the coordinate representation, not the existence of the Cartesian point.
A point with x=0 but y nonzero is not singular. It lies on a positive or negative y-axis ray and has azimuth 90 or 270 degrees. A point with y=0 but x nonzero likewise has zero or 180 degrees. Testing those near-axis cases is useful because a careless implementation can treat one zero coordinate as if both were zero. The singular condition is the pair, not either field alone.
- The entire z-axis has undefined azimuth.
- z does not resolve the planar direction.
- One zero planar coordinate is valid when the other is nonzero.
- The handler rejects the singular pair before angle formatting.
Cylindrical versus spherical coordinates
Cylindrical rho measures only distance in the xy-plane, while spherical rho measures distance from the three-dimensional origin. Cylindrical notation uses one azimuth and z; spherical notation adds a polar angle and uses the z contribution in the radial distance. A point such as (1,1,3) makes the distinction clear: cylindrical rho is sqrt(2), whereas the spherical radial distance is sqrt(11). Both descriptions can be correct because they answer different coordinate questions.
This page is also different from a polar two-dimensional conversion because it retains an axial z field. It does not calculate a velocity component, a force direction, or a rotating-frame quantity merely because those applications often use cylindrical notation. The coordinate conversion is a reusable arithmetic layer; any physical interpretation requires a stated origin, axis, units, and time or field model outside this handler.
- Cylindrical rho excludes z from its distance.
- Spherical coordinates use a 3D radial distance and polar angle.
- Cylindrical coordinates retain an axial z component.
- Notation alone does not identify a physical quantity.
Validation and renderer-safe output
The handler checks all three inputs for finite type and range before computing rho. It checks the radial distance, raw azimuth, and normalized azimuth, and it normalizes negative zero in numeric entries. The output has three finite numeric results with units that distinguish planar length, degrees, and axial length. No angle sentinel is used for the singular axis; the explicit error is safer than allowing an ambiguous value into later work.
The steps expose the square-root expression, atan2 convention, normalization range, and unchanged z. A renderer can format the entries without parsing an angle string or a coordinate tuple. If a caller needs radians, it should convert the finite degree result with a separately documented rule. The handler does not round rho or z before return, so display formatting remains separate from numeric reuse.
- All inputs and derived values are finite-checked.
- The z-axis error is model-specific and explicit.
- rho, azimuth, and z have distinct labels and units.
- Numeric values are returned without presentation rounding.
Source boundary, assumptions, and conservative limits
The cylindrical-coordinate reference stored in the catalog is private formula metadata used for review. This public article and its examples are original WorldCalculate writing. The reference supports the standard coordinate definitions but does not supply another site's page body, code, branding, defaults, or calculator data. It also cannot establish the frame or units of a visitor's coordinates. Those details remain explicit assumptions of the person preparing the inputs.
The model assumes a flat Cartesian system, a positive orientation convention around the z-axis, finite values, and degree output. It excludes coordinate-frame transformations, uncertainty, curved spaces, and time-dependent axes. Do not use a finite triple as proof of a physical orientation, navigation solution, or equipment state without validating the underlying frame and measurement process. The conservative limit is conversion only: one point description goes in and one equivalent description comes out.
- Private metadata provides formula provenance only.
- The axis orientation and units must be known before entry.
- Curved, moving, or uncertain frames are not modeled.
- The conversion is not navigation or physical-state validation.
Frequently asked questions
Why is azimuth undefined when rho is zero? At the z-axis there is no nonzero planar vector and therefore no direction from the positive x-axis. Why is azimuth sometimes 0 rather than 360? The half-open convention chooses one representative of the same direction. Why does z stay unchanged? Cylindrical coordinates change the description of the xy position while using the same axial coordinate. Why not include z in rho? Including it would create the spherical radial distance and answer a different question.
Can negative rho be returned? No. rho is a distance and is computed as a nonnegative square root. Could the result be in radians? This record explicitly chooses degrees; convert outside if another model needs radians. Are the bounds mathematical? No. They are conservative software limits for finite arithmetic. Keep coordinate units, origin, axis orientation, and any uncertainty with the result before using it in a consequential calculation.
- rho=0 removes the planar direction needed for azimuth.
- 360 degrees is represented by 0 under the chosen range.
- z is unchanged because it is the axial coordinate.
- Radians, frame transforms, and physical decisions are outside this contract.
Reconstructing the planar coordinates
For a non-singular output, the planar coordinates can be reconstructed conceptually as x=rho*cos(phi) and y=rho*sin(phi), after converting the reported degree angle to radians. This relationship explains why rho and azimuth together retain the xy position. The current handler does not perform the inverse conversion and does not return sine or cosine components. Keeping conversion directions separate avoids a caller confusing a forward Cartesian-to-cylindrical result with an inverse result.
The axial coordinate needs no reconstruction: cylindrical z is exactly the input z. A forward result with rho positive and an angle in the selected range corresponds to one planar direction, while rho zero has no unique angle. If a downstream system receives a singular-axis point, it should preserve an explicit undefined angle rather than assign a value and claim reversibility. The coordinate tuple is only equivalent when its frame and conventions match.
- rho and azimuth jointly describe x and y.
- Inverse reconstruction needs an explicit degree-to-radian step.
- z is already the axial coordinate.
- The singular axis cannot be inverted to one angle.
Axis points and negative-zero behavior
The four planar axes are useful branch checks. (positive x,0) maps to azimuth 0, (0, positive y) maps to 90, (negative x,0) maps to 180, and (0, negative y) maps to 270. Their rho values are the absolute nonzero coordinate. These cases verify that atan2 receives the arguments in the order atan2(y, x) and that the normalization does not accidentally swap axes or report a negative angle.
Signed zero can appear in JavaScript's floating-point representation even when a coordinate is mathematically zero. The shared numeric helper normalizes negative zero in returned values, and the angle branch folds an endpoint at 360 to zero. This makes output comparisons and serialized results stable. It does not erase a meaningful negative coordinate; only the representation of zero is normalized.
- Positive x is azimuth 0 degrees.
- Positive y is 90 degrees and negative y is 270 degrees.
- Negative x is 180 degrees.
- Negative zero is normalized without changing signed nonzero coordinates.
Angle convention and interoperability
Different systems may measure azimuth from another axis, rotate clockwise, or use radians. This record is explicit: the reference ray is positive x, positive angular direction follows the usual atan2 convention, and the output is degrees in [0,360). A receiving system must convert deliberately if it uses a different convention. Copying the numeric angle without copying the convention can move a point to another quadrant even though the number appears valid.
The degree range is half-open so zero and 360 are not duplicate representations. An angle of 359.9 is valid and remains close to the positive x-axis from the negative-y side. A negative raw atan2 value is shifted once by 360. The handler does not accept an angle field because the angle is derived from x and y; adding one would create an inconsistent over-specified point.
- The reference ray is positive x.
- Positive direction follows atan2(y, x).
- Output is degrees in [0,360).
- Receiving systems must preserve or convert the convention explicitly.
Frame, units, and measured coordinates
Cylindrical conversion is coordinate-frame dependent. Translating the origin or rotating the x-y axes changes rho and azimuth even when the physical point is unchanged. The calculator assumes the input frame already has the desired z-axis and origin. It does not apply a datum transformation, local-to-global rotation, map projection, or sensor calibration. Those operations should be completed and documented before the three fields reach this handler.
The rho and z outputs use the length unit of their respective input coordinates, while azimuth has degrees. A mixed unit pair in x and y changes the radial and angular meaning, not just the label. Scaling x, y, and z by a common factor scales rho and z but leaves azimuth unchanged. These relationships are helpful checks for transformed data, but they do not validate a physical frame or measurement uncertainty.
- Origin and axes are assumed known before conversion.
- No frame transformation or calibration is performed.
- rho and z retain coordinate length units.
- Common length scaling leaves azimuth unchanged.
Focused checks for coordinate conversion
A focused test should use (1,1,3) for rho=sqrt(2), azimuth 45, and z=3; use all four axis rays to verify quadrant normalization; and use a negative z to verify that the planar outputs do not change. It should reject (0,0,0) and (0,0, nonzero) with the exact undefined-azimuth error. It should also exercise maximum finite coordinates and direct invalid values so the singular and numeric boundaries are both covered.
The catalog test should verify field order x, y, z, the degree unit, the [0,360) wording, and the example keys. The generic registry test should invoke the same handler from the merged map and check that its three results are finite, serializable, and labeled. These checks matter because a coordinate conversion can return plausible numbers after an argument swap; known quadrant answers catch that defect more reliably than a default case alone.
- Use a first-quadrant known answer.
- Exercise all four cardinal azimuths.
- Reject the entire z-axis singular case.
- Test the registered handler, not only a direct import.
A reversible description with an explicit singularity
For ordinary points, the three returned values preserve enough information to describe the original Cartesian point after an explicitly documented inverse conversion. That reversibility depends on keeping the degree convention, positive-x reference ray, and common units with the result. It does not mean every tuple has a unique inverse: all points on the z-axis share rho=0 and differ only in z, while their azimuths remain undefined. The handler makes that exception visible instead of placing a false value into an otherwise reversible representation.
This boundary is useful in applications that use radial bins or angular sectors. A point near the axis may have a valid but highly sensitive angle because small x and y changes can move it across sectors. The calculator reports the exact finite angle for the entered point but does not attach an uncertainty cone or sector decision. If classification near the axis matters, carry measurement error and a tolerance policy into the downstream model.
- Ordinary tuples can be inverted with the same convention.
- The z-axis has no unique azimuth.
- Near-axis angles can be sensitive to small planar changes.
- Sector classification needs an external uncertainty policy.
Coordinate conversion handoff
When a cylindrical result is passed to another calculation, include the original Cartesian point and the convention used to derive rho and azimuth. This makes it possible to verify the conversion and prevents an angle in degrees from being mistaken for radians. Preserve the distinction between a valid ordinary point and the explicit z-axis undefined-azimuth case. A downstream radial or angular binning step should define its own boundary behavior rather than relying on display rounding.
The three outputs are descriptive coordinates, not a physical state. A field, velocity, force, navigation bearing, or rotating-frame quantity needs a separate model with its own units and frame transformations. The calculator's finite result confirms only that one Cartesian-to-cylindrical conversion completed inside the declared bounds. Retaining the frame, axis direction, origin, and source precision is part of using the result responsibly.
- Carry Cartesian inputs with cylindrical outputs.
- Preserve degrees and the [0,360) convention.
- Keep undefined azimuth explicit on the z-axis.
- Physical vector interpretation requires another model.
Frequently asked questions
What is the Cylindrical Coordinates?
Convert Cartesian x, y, z coordinates to cylindrical rho, normalized azimuth, and unchanged z.
What is the formula for the Cylindrical Coordinates?
rho=sqrt(x^2+y^2), azimuth=atan2(y,x) normalized to [0,360) degrees, and z_cylindrical=z; azimuth is undefined when x=y=0. Cylindrical coordinates measure distance from the z-axis in the xy-plane, angle from the positive x-axis, and height along z. The azimuth is normalized to a half-open degree range and the z value is unchanged.
What do I need to use this calculator?
Enter Cartesian x, Cartesian y, Cartesian z, then choose Calculate.
What are the limits of this calculator?
The coordinates are finite points in one right-handed Cartesian system with compatible units. Azimuth is reported in degrees from 0 inclusive through 360 exclusive, and the positive z-axis is represented by rho=0 with no defined azimuth. This is a coordinate conversion only; it does not calculate spherical polar angle, velocity, or a physical orientation.
Methodology
This calculator is part of the WorldCalculate library. Its formula, example, assumptions, input bounds, and output formatting follow the official methodology.
Use this calculator as part of a bigger plan
These WorldCalculate collections connect this tool with related questions while keeping each calculation separate and transparent.