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Math · Complex Numbers and Parametric Motion

Parametric Curves / Motion from Equations

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Wrap-up

What you learned

Recommended next: Open concept testCheck whether the core ideas are ready without leaving this concept.
Read next: Vectors in 2DUse the same plane and ordered-pair language to read direction and displacement.

Key takeaway

A shared parameter t selects one ordered pair (x(t), y(t)) at a time.
Amplitudes mainly change the horizontal and vertical reach of the path.
Frequencies, phase shift, and the current time explain how the point travels through the trace.
The full path and the moving-point/time readouts answer different questions, so both views matter.

Common misconception

Do not treat the picture of the curve as the whole story. The same-looking path can hide a different traversal, and a timing change can alter the shape too.

The traced path and the timing along that path are related but different ideas.

Keep this idea moving

Open the next concept, route, or track only when you want the current model to widen into a larger branch.

Carry the same plane language into vectors and directionVectors

Vectors in 2D

Combine, subtract, and scale vectors on one plane so magnitude, direction, and components stay tied to the same live object.

Next in Trig geometry on the planeComplex Numbers and Parametric Motion

Trig Identities from Unit-Circle Geometry

Keep one rotating point and its projections visible so the core trig identities stay tied to geometry instead of detached symbol rules.

Next in Angle recovery from x and yComplex Numbers and Parametric Motion

Inverse Trig / Angle from Ratio

Keep one polar point and its coordinate signs visible so inverse trig becomes angle-from-ratio reasoning with quadrant checks instead of a calculator-only output.

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More practice optionsReview on the bench · Worked examples

Practice optionReview on the benchReplay the guided setup with the key controls and graphs in sight.
Practice optionWorked examplesCompare against solved walkthroughs after you try the setup yourself.
Practice optionFree play on the benchExperiment freely with the live controls before moving to another concept.

Reference and support

Use these calmer sections when you want a fuller explanation, examples, or accessibility support after the guided flow.

Return here when you want the slower reference pass.

Open reference and support

Equation snapshotHorizontal coordinate rule · Vertical coordinate ruleShow 3 equations

Read x(t) and y(t) together as one ordered pair, then use the speed rule only when you need the traversal story.

Horizontal coordinate rule
$x (t) = A_{x} cos (n_{x} t)$
Gives the x-coordinate selected by the current time t.
Vertical coordinate rule
$y (t) = A_{y} sin (n_{y} t + ϕ)$
Gives the y-coordinate selected by the same t, with an optional phase offset.
Speed along the curve
$v = (\frac{d x}{d t})^{2} + (\frac{d y}{d t})^{2}$
Measures how fast the point moves through the traced path, which is a motion question rather than a shape question.

Why it behaves this way

Explanation

A parametric curve is built by one shared clock t. At each moment, x(t) and y(t) together choose one point in the plane. This bench keeps those two coordinate values, the moving point, and the traced path visible at once so the equations stay tied to the geometry.

Keep two questions separate. First, what set of points gets traced out? That is the path. Second, how does the point move through those points as time passes? That is the motion. Amplitudes mainly set the size of the path, while frequencies and phase shift control how the point is timed along it.

Key ideas

01At each time t, x(t) and y(t) must be read together as one ordered pair (x(t), y(t)).

02Amplitudes set the horizontal and vertical reach of the traced path.

03Frequencies and phase shift change the timing between x and y, so they affect how the point moves through the curve and can also change the shape that gets traced.

Worked examples

Open examples when you want to see the same idea walked through step by step.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use the live time slider and read x(t) and y(t) as one ordered pair. The full trace shows where the point can go, and the current time tells you which point is active now.

Supporter unlocks saved study tools, exact-state sharing, and the richer review surfaces that support this guided flow.

View plans

Frozen valuesUsing frozen parameters

At the current time, what ordered pair do x(t) and y(t) assign, and where is that point on the traced path?

A_{x}

x amplitude

3.2

A_{y}

y amplitude

2.4

ϕ

Phase shift

1. Read the current time

The current time is t = 0.

2. Read x(t) and y(t) as one pair

At this moment, x(t) = 3.2 and y(t) = 0.

3. Place that pair on the plane

So the moving point is at (3.2, 0) on the traced curve.

Current position on the path

(x, y) = (3.2, 0)

The point is moving more slowly here even though the traced path itself has not changed.

Quick test

Loading saved test state.

Accessibility

Open the text-first descriptions when you need the simulation and graph translated into words.

The simulation shows a parametric curve, a moving point, and controls for amplitudes, frequencies, and phase shift. The point moves through the same plane where the whole path is traced.

Keeping the full trace visible shows the geometric path, while the moving point and time trail show how that path is traversed over time.

Graph summary

One graph shows x(t) and y(t) together, so each time corresponds to one ordered pair on the plane.

A second graph shows the point's speed over time, which helps separate the shape of the path from how fast the point moves along it.

Bench tools and share links

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Current bench

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Featured setups

Ellipse with full trace

Open a same-frequency ellipse with the whole path and recent trail visible.

Phase-shifted loop

Open a loop with a phase offset and compare the full trace with the moving point.

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Progress

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Starter track

Step 6 of 6

Complex and Parametric Motion

Parametric Curves / Motion from Equations appears later in this track, so it is cleaner to start from the beginning first.

Previous step: Inverse Trig / Angle from Ratio

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