HomeConcept libraryMathComplex Numbers and Parametric MotionComplex and Parametric Motion

Math Complex Numbers and Parametric MotionIntroStarter track

Concept module

Parametric Curves / Motion from Equations

Keep x(t), y(t), the traced path, and the moving point visible together so shape and traversal stay distinct.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Time

0.00 s / 6.28 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s6.28 s

Parametric curves and motion

Keep x(t), y(t), the traced path, and the moving point tied together so shape and traversal do not collapse into the same idea.

Graphs

Switch graph views without breaking the live stage and time link.

Coordinates vs time

One graph shows x(t) and y(t) together, and a second graph shows the point's speed over time.

time: 0 to 6.28coordinate value: -4 to 4

x(t)y(t)

Hover or scrub to link the graph back to the stage.time / coordinate value

Controls

Adjust the live parameters and watch the bench respond.

x amplitude

A_{x}

3.2

y amplitude

A_{y}

2.4

x frequency

n_{x}

y frequency

n_{y}

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

Phase shift

ϕ

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

ObservationPrompt 1 of 2

Changing amplitudes changes the width and height of the curve even if the timing stays the same.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

A_{x}

x amplitude

3.2

Sets how far the path stretches horizontally.

Graph: Coordinates vs timeOverlay: Full pathOverlay: Coordinate projections

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Keep the traced curve and the moving point visible together.

ObservationPrompt 1 of 2

Graph: Coordinates vs time

Changing amplitudes changes the width and height of the curve even if the timing stays the same.

Control: x amplitudeControl: y amplitudeGraph: Coordinates vs timeOverlay: Full pathEquation

A_{x}

Equation

A_{y}

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

2 visible

Overlay focus

Full path

Keep the complete traced curve visible.

What to notice

The shape of the curve is easier to compare when the whole trace stays in view.

Why it matters

It separates the path from the moving point.

Control: x amplitudeControl: y amplitudeGraph: Coordinates vs timeEquation

A_{x}

Equation

A_{y}

Challenge mode

Build the shape first, then respect the timing of the moving point.

0/1 solved

ConditionCore

3 of 8 checks

Tall, fast, and near the axis

Build a curve that is clearly taller than it is wide, then pause when the point is near the y-axis and still moving quickly.

Graph-linkedGuided start

Pending

Open the Speed vs time graph.

Coordinates vs time

Matched

Keep the Full path visible.

Matched

Keep the Time trail visible.

Matched

Return to live time.

live

Pending

Keep path height between 6.4 and 8.5.

4.8

Pending

Keep path width between 2 and 5.4.

6.4

Pending

Keep horizontal position between -0.35 and 0.35.

3.2

Pending

Keep speed between 6.5 and 20.

4.8

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

At t = 0, the point is near (3.2, 0). The path spans about 6.4 units wide and 4.8 units tall. The point is moving at a moderate speed through the traced path.

Equation detailsDeeper interpretation, notes, and worked variable context.

x-coordinate rule

Sets the horizontal motion as a function of the parameter t.

x (t) = A_{x} cos (n_{x} t)

A_{x}

x amplitude 3.2

n_{x}

x frequency 1

y-coordinate rule

Sets the vertical motion and allows a phase offset.

y (t) = A_{y} sin (n_{y} t + ϕ)

A_{y}

y amplitude 2.4

n_{y}

y frequency 2

ϕ

Phase shift 0

Speed along the path

Separates the point's motion from the path it happens to trace.

v = (\frac{d x}{d t})^{2} + (\frac{d y}{d t})^{2}

n_{x}

x frequency 1

n_{y}

y frequency 2

Progress

Not startedMastery: NewLocal-first

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Let the live model runChange one real controlOpen What to notice

Try this setup

Jump to a named bench state or copy the one you are looking at now. Shared links reopen the same controls, graph, overlays, and compare context.

Featured setups

Ellipse trace

Open a one-frequency ellipse with the full path and time trail visible.

Phase-shift loop

Open a phase-shifted loop and compare the traced path with the moving point.

Saved setups

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Checking saved setup access.

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Copy current setup

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Stable links

Starter track

Step 2 of 20 / 2 complete

Complex and Parametric Motion

Earlier steps still set up Parametric Curves / Motion from Equations.

1. Complex Numbers on the Plane2. Parametric Curves / Motion from Equations

Previous step: Complex Numbers on the Plane.

Start from track beginning

Short explanation

What the system is doing

Parametric curves become easier to trust when the path and the motion along that path stay visible together. This bench keeps x(t), y(t), the traced curve, and the moving point tied to the same time slider.

The goal is to separate two ideas that often blur together: the shape traced out in the plane and the timing of how the point moves through that shape.

Key ideas

01x(t) and y(t) work together to determine one moving point in the plane.

02Changing amplitudes changes the size of the traced path, while changing frequencies changes how the point traverses it.

03A path can stay the same overall shape while the point speeds up and slows down along it.

Worked example

Read the full frozen walkthrough.

Frozen walkthrough

Use the current parametric bench instead of a detached substitution problem.

Live worked examples are available on Premium. You can still read the full frozen walkthrough on the free tier.

View plans

Frozen valuesUsing frozen parameters

At the current time, where is the point on the parametric curve?

A_{x}

x amplitude

3.2

A_{y}

y amplitude

2.4

ϕ

Phase shift

1. Start from the current time

The current time is t = 0.

2. Read x(t) and y(t) together

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 point

(x, y) = (3.2, 0)

The point is moving at a moderate speed here, so the curve and the motion cue still feel tightly linked.

Common misconception

If the curve looks the same, the motion along it must also be the same.

The traced path and the time-progress along it are related but not identical ideas.

The same kind of curve can be traversed at different speeds or with different timing between x and y.

Mini challenge

Build a path that is taller than it is wide, then catch the point near the y-axis while it is still moving quickly.

Prediction prompt

Decide whether amplitudes or frequencies control the shape and which settings mostly affect the speed.

Check your reasoning

The amplitudes set the width and height, while the frequencies and current time help determine how fast the point is moving at a given place.

That is why the same idea has both a path shape and a traversal story.

Quick test

Misconception check

Question 1 of 2

Answer from the live curve and motion cues.

Which statement best separates the curve from the motion along it?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

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.

Graph summary

One graph shows x(t) and y(t) together, and a second graph shows the point's speed over time.

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Carry the same plane language into vectors and directionVectors

Parametric Curves / Motion from Equations

See each variable before you move it.

Full path

Tall, fast, and near the axis

Complex and Parametric Motion

What the system is doing

Read the full frozen walkthrough.

At the current time, where is the point on the parametric curve?

Which statement best separates the curve from the motion along it?

Keep moving through the concept map.

Vectors in 2D

Complex Numbers on the Plane

Projectile Motion