Curvilinear coordinates

Cartesian coordinates use fixed horizontal and vertical axes. Polar coordinates describe the same point by distance from an origin and an angle around it, which is often clearer for circular motion.

Think like a programmer

Coordinate systems are encoding choices. Convert at the boundary of an algorithm, preserve the chosen coordinate type in names, and never mix an angle with a length just because both are numbers.

Model checklist

Inputs: Radius in metres and angle in radians.
State: One polar coordinate value.
Rule: Apply sine and cosine to convert to Cartesian components.
Output: Equivalent x and y position.
Check: Conversion to Cartesian and back preserves radius.

\[x=r\\cos\\theta,\\qquad y=r\\sin\\theta\]

Cartesian and polar descriptions

Set a radius and an angle. The same position appears as polar input and Cartesian output.

Radius: 3.0 mAngle: 45°

r = 3.0 m, θ = 45°; Cartesian position [2.12, 2.12] m.

Try this experiment

Prediction: Changing angle changes Cartesian components while leaving radius fixed.

Hold radius at 3 m and rotate through 90°. Watch the text coordinates and identify the moment where x becomes zero.

Where this model breaks

Polar coordinates are singular at radius zero because the angle is undefined. Cylindrical and spherical coordinates add dimensions but introduce similar poles and basis changes.

Summary

Choose coordinates that simplify the question. Convert deliberately, keep units visible, and test a round trip between representations.

Glossary

Cartesian coordinates: components along fixed perpendicular axes.
Polar coordinates: radius and angle in a plane.
Singularity: a representation point where part of the coordinate data is undefined.

Self-check

What are the units of radius and angle?
What happens to x at 90°?
Where is polar angle undefined?

Sources

OpenStax, University Physics Volume 1, coordinate systems chapter.

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Curvilinear Coordinates, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Motion in 2D and 3D, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Curvilinear Coordinates as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Curvilinear Coordinates into a test

Convert a point between Cartesian and polar coordinates and learn why components depend on the coordinate system.

Name the inputs and units that the motion in 2d and 3d model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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