Parametric Polar Cosine Curve: r = cos(a/b · t)

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2024-04-08 17:41

In collections intersection: surface-surface, curve-curve

This example demonstrates how to plot the polar curve r = cos(a/b · t) and visualize its periodic structure and amplitude behavior as the parameter ratio a/b changes. The key mathematical ideas are polar-coordinate representation, trigonometric functions (cosine), parameter-driven motion via the variable t, and the continuity implied by the underlying trigonometric expression, which yields a smooth drawn locus. In GeoGebra, the construction uses Sequence to generate a sampled parameter list and CurveCartesian to connect the resulting parameterized points into a single curve. Adjustable objects such as a and b (and auxiliary parameters like c) control the frequency and overall shape, while Text and related display objects (e.g., Text1, Tw, Ts, Text2) help present the formula and parameter meanings during the construction. Suitable for high school through early university learners studying trigonometry, parametric/graphing methods, and interactive visualization.

GeoGebra Commands & Parameters

Sequence(Sequence[CurveCartesian[(2 * j) + (cos(t) * cos((j / k * t))), (2 * k) + (sin(t) * cos((j / k * t))), t, 0, (20 * pi)], j, 1, 10], k, 1, 10) → K, {{((2 * 1) + (cos(t) * cos((1 / 1 * t))), (2 * 1) + (sin(t) * cos((1 / 1 * t)))), ((2 * 2) + (cos(t) * cos((2 / 1 * t))), (2 * 1) + (sin(t) * cos((2 / 1 * t)))), ((2 * 3) + (cos(t) * cos((3 / 1 * t))), (2 * 1) + (sin(t) * cos((3 / 1 * t)))), ((2 * 4) + (cos(t) * cos((4 / 1 * t))), (2 * 1) + (sin(t) * cos((4 / 1 * t)))), ((2 * 5) + (cos(t) * cos((5 / 1 * t))), (2 * 1) + (sin(t) * cos((5 / 1 * t)))), ((2 * 6) + (cos(t) * cos((6 / 1 * t))), (2 * 1) + (sin(t) * cos((6 / 1 * t)))), ((2 * 7) + (cos(t) * cos((7 / 1 * t))), (2 * 1) + (sin(t) * cos((7 / 1 * t)))), ((2 * 8) + (cos(t) * cos((8 / 1 * t))), (2 * 1) + (sin(t) * cos((8 / 1 * t)))), ((2 * 9) + (cos(t) * cos((9 / 1 * t))), (2 * 1) + (sin(t) * cos((9 / 1 * t)))), ((2 * 10) + (cos(t) * cos((10 / 1 * t))), (2 * 1) + (sin(t) * cos((10 / 1 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 2 * t))), (2 * 2) + (sin(t) * cos((1 / 2 * t)))), ((2 * 2) + (cos(t) * cos((2 / 2 * t))), (2 * 2) + (sin(t) * cos((2 / 2 * t)))), ((2 * 3) + (cos(t) * cos((3 / 2 * t))), (2 * 2) + (sin(t) * cos((3 / 2 * t)))), ((2 * 4) + (cos(t) * cos((4 / 2 * t))), (2 * 2) + (sin(t) * cos((4 / 2 * t)))), ((2 * 5) + (cos(t) * cos((5 / 2 * t))), (2 * 2) + (sin(t) * cos((5 / 2 * t)))), ((2 * 6) + (cos(t) * cos((6 / 2 * t))), (2 * 2) + (sin(t) * cos((6 / 2 * t)))), ((2 * 7) + (cos(t) * cos((7 / 2 * t))), (2 * 2) + (sin(t) * cos((7 / 2 * t)))), ((2 * 8) + (cos(t) * cos((8 / 2 * t))), (2 * 2) + (sin(t) * cos((8 / 2 * t)))), ((2 * 9) + (cos(t) * cos((9 / 2 * t))), (2 * 2) + (sin(t) * cos((9 / 2 * t)))), ((2 * 10) + (cos(t) * cos((10 / 2 * t))), (2 * 2) + (sin(t) * cos((10 / 2 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 3 * t))), (2 * 3) + (sin(t) * cos((1 / 3 * t)))), ((2 * 2) + (cos(t) * cos((2 / 3 * t))), (2 * 3) + (sin(t) * cos((2 / 3 * t)))), ((2 * 3) + (cos(t) * cos((3 / 3 * t))), (2 * 3) + (sin(t) * cos((3 / 3 * t)))), ((2 * 4) + (cos(t) * cos((4 / 3 * t))), (2 * 3) + (sin(t) * cos((4 / 3 * t)))), ((2 * 5) + (cos(t) * cos((5 / 3 * t))), (2 * 3) + (sin(t) * cos((5 / 3 * t)))), ((2 * 6) + (cos(t) * cos((6 / 3 * t))), (2 * 3) + (sin(t) * cos((6 / 3 * t)))), ((2 * 7) + (cos(t) * cos((7 / 3 * t))), (2 * 3) + (sin(t) * cos((7 / 3 * t)))), ((2 * 8) + (cos(t) * cos((8 / 3 * t))), (2 * 3) + (sin(t) * cos((8 / 3 * t)))), ((2 * 9) + (cos(t) * cos((9 / 3 * t))), (2 * 3) + (sin(t) * cos((9 / 3 * t)))), ((2 * 10) + (cos(t) * cos((10 / 3 * t))), (2 * 3) + (sin(t) * cos((10 / 3 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 4 * t))), (2 * 4) + (sin(t) * cos((1 / 4 * t)))), ((2 * 2) + (cos(t) * cos((2 / 4 * t))), (2 * 4) + (sin(t) * cos((2 / 4 * t)))), ((2 * 3) + (cos(t) * cos((3 / 4 * t))), (2 * 4) + (sin(t) * cos((3 / 4 * t)))), ((2 * 4) + (cos(t) * cos((4 / 4 * t))), (2 * 4) + (sin(t) * cos((4 / 4 * t)))), ((2 * 5) + (cos(t) * cos((5 / 4 * t))), (2 * 4) + (sin(t) * cos((5 / 4 * t)))), ((2 * 6) + (cos(t) * cos((6 / 4 * t))), (2 * 4) + (sin(t) * cos((6 / 4 * t)))), ((2 * 7) + (cos(t) * cos((7 / 4 * t))), (2 * 4) + (sin(t) * cos((7 / 4 * t)))), ((2 * 8) + (cos(t) * cos((8 / 4 * t))), (2 * 4) + (sin(t) * cos((8 / 4 * t)))), ((2 * 9) + (cos(t) * cos((9 / 4 * t))), (2 * 4) + (sin(t) * cos((9 / 4 * t)))), ((2 * 10) + (cos(t) * cos((10 / 4 * t))), (2 * 4) + (sin(t) * cos((10 / 4 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 5 * t))), (2 * 5) + (sin(t) * cos((1 / 5 * t)))), ((2 * 2) + (cos(t) * cos((2 / 5 * t))), (2 * 5) + (sin(t) * cos((2 / 5 * t)))), ((2 * 3) + (cos(t) * cos((3 / 5 * t))), (2 * 5) + (sin(t) * cos((3 / 5 * t)))), ((2 * 4) + (cos(t) * cos((4 / 5 * t))), (2 * 5) + (sin(t) * cos((4 / 5 * t)))), ((2 * 5) + (cos(t) * cos((5 / 5 * t))), (2 * 5) + (sin(t) * cos((5 / 5 * t)))), ((2 * 6) + (cos(t) * cos((6 / 5 * t))), (2 * 5) + (sin(t) * cos((6 / 5 * t)))), ((2 * 7) + (cos(t) * cos((7 / 5 * t))), (2 * 5) + (sin(t) * cos((7 / 5 * t)))), ((2 * 8) + (cos(t) * cos((8 / 5 * t))), (2 * 5) + (sin(t) * cos((8 / 5 * t)))), ((2 * 9) + (cos(t) * cos((9 / 5 * t))), (2 * 5) + (sin(t) * cos((9 / 5 * t)))), ((2 * 10) + (cos(t) * cos((10 / 5 * t))), (2 * 5) + (sin(t) * cos((10 / 5 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 6 * t))), (2 * 6) + (sin(t) * cos((1 / 6 * t)))), ((2 * 2) + (cos(t) * cos((2 / 6 * t))), (2 * 6) + (sin(t) * cos((2 / 6 * t)))), ((2 * 3) + (cos(t) * cos((3 / 6 * t))), (2 * 6) + (sin(t) * cos((3 / 6 * t)))), ((2 * 4) + (cos(t) * cos((4 / 6 * t))), (2 * 6) + (sin(t) * cos((4 / 6 * t)))), ((2 * 5) + (cos(t) * cos((5 / 6 * t))), (2 * 6) + (sin(t) * cos((5 / 6 * t)))), ((2 * 6) + (cos(t) * cos((6 / 6 * t))), (2 * 6) + (sin(t) * cos((6 / 6 * t)))), ((2 * 7) + (cos(t) * cos((7 / 6 * t))), (2 * 6) + (sin(t) * cos((7 / 6 * t)))), ((2 * 8) + (cos(t) * cos((8 / 6 * t))), (2 * 6) + (sin(t) * cos((8 / 6 * t)))), ((2 * 9) + (cos(t) * cos((9 / 6 * t))), (2 * 6) + (sin(t) * cos((9 / 6 * t)))), ((2 * 10) + (cos(t) * cos((10 / 6 * t))), (2 * 6) + (sin(t) * cos((10 / 6 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 7 * t))), (2 * 7) + (sin(t) * cos((1 / 7 * t)))), ((2 * 2) + (cos(t) * cos((2 / 7 * t))), (2 * 7) + (sin(t) * cos((2 / 7 * t)))), ((2 * 3) + (cos(t) * cos((3 / 7 * t))), (2 * 7) + (sin(t) * cos((3 / 7 * t)))), ((2 * 4) + (cos(t) * cos((4 / 7 * t))), (2 * 7) + (sin(t) * cos((4 / 7 * t)))), ((2 * 5) + (cos(t) * cos((5 / 7 * t))), (2 * 7) + (sin(t) * cos((5 / 7 * t)))), ((2 * 6) + (cos(t) * cos((6 / 7 * t))), (2 * 7) + (sin(t) * cos((6 / 7 * t)))), ((2 * 7) + (cos(t) * cos((7 / 7 * t))), (2 * 7) + (sin(t) * cos((7 / 7 * t)))), ((2 * 8) + (cos(t) * cos((8 / 7 * t))), (2 * 7) + (sin(t) * cos((8 / 7 * t)))), ((2 * 9) + (cos(t) * cos((9 / 7 * t))), (2 * 7) + (sin(t) * cos((9 / 7 * t)))), ((2 * 10) + (cos(t) * cos((10 / 7 * t))), (2 * 7) + (sin(t) * cos((10 / 7 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 8 * t))), (2 * 8) + (sin(t) * cos((1 / 8 * t)))), ((2 * 2) + (cos(t) * cos((2 / 8 * t))), (2 * 8) + (sin(t) * cos((2 / 8 * t)))), ((2 * 3) + (cos(t) * cos((3 / 8 * t))), (2 * 8) + (sin(t) * cos((3 / 8 * t)))), ((2 * 4) + (cos(t) * cos((4 / 8 * t))), (2 * 8) + (sin(t) * cos((4 / 8 * t)))), ((2 * 5) + (cos(t) * cos((5 / 8 * t))), (2 * 8) + (sin(t) * cos((5 / 8 * t)))), ((2 * 6) + (cos(t) * cos((6 / 8 * t))), (2 * 8) + (sin(t) * cos((6 / 8 * t)))), ((2 * 7) + (cos(t) * cos((7 / 8 * t))), (2 * 8) + (sin(t) * cos((7 / 8 * t)))), ((2 * 8) + (cos(t) * cos((8 / 8 * t))), (2 * 8) + (sin(t) * cos((8 / 8 * t)))), ((2 * 9) + (cos(t) * cos((9 / 8 * t))), (2 * 8) + (sin(t) * cos((9 / 8 * t)))), ((2 * 10) + (cos(t) * cos((10 / 8 * t))), (2 * 8) + (sin(t) * cos((10 / 8 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 9 * t))), (2 * 9) + (sin(t) * cos((1 / 9 * t)))), ((2 * 2) + (cos(t) * cos((2 / 9 * t))), (2 * 9) + (sin(t) * cos((2 / 9 * t)))), ((2 * 3) + (cos(t) * cos((3 / 9 * t))), (2 * 9) + (sin(t) * cos((3 / 9 * t)))), ((2 * 4) + (cos(t) * cos((4 / 9 * t))), (2 * 9) + (sin(t) * cos((4 / 9 * t)))), ((2 * 5) + (cos(t) * cos((5 / 9 * t))), (2 * 9) + (sin(t) * cos((5 / 9 * t)))), ((2 * 6) + (cos(t) * cos((6 / 9 * t))), (2 * 9) + (sin(t) * cos((6 / 9 * t)))), ((2 * 7) + (cos(t) * cos((7 / 9 * t))), (2 * 9) + (sin(t) * cos((7 / 9 * t)))), ((2 * 8) + (cos(t) * cos((8 / 9 * t))), (2 * 9) + (sin(t) * cos((8 / 9 * t)))), ((2 * 9) + (cos(t) * cos((9 / 9 * t))), (2 * 9) + (sin(t) * cos((9 / 9 * t)))), ((2 * 10) + (cos(t) * cos((10 / 9 * t))), (2 * 9) + (sin(t) * cos((10 / 9 * t))))}, {((2 * 1) + (cos(t) * cos((1 / 10 * t))), (2 * 10) + (sin(t) * cos((1 / 10 * t)))), ((2 * 2) + (cos(t) * cos((2 / 10 * t))), (2 * 10) + (sin(t) * cos((2 / 10 * t)))), ((2 * 3) + (cos(t) * cos((3 / 10 * t))), (2 * 10) + (sin(t) * cos((3 / 10 * t)))), ((2 * 4) + (cos(t) * cos((4 / 10 * t))), (2 * 10) + (sin(t) * cos((4 / 10 * t)))), ((2 * 5) + (cos(t) * cos((5 / 10 * t))), (2 * 10) + (sin(t) * cos((5 / 10 * t)))), ((2 * 6) + (cos(t) * cos((6 / 10 * t))), (2 * 10) + (sin(t) * cos((6 / 10 * t)))), ((2 * 7) + (cos(t) * cos((7 / 10 * t))), (2 * 10) + (sin(t) * cos((7 / 10 * t)))), ((2 * 8) + (cos(t) * cos((8 / 10 * t))), (2 * 10) + (sin(t) * cos((8 / 10 * t)))), ((2 * 9) + (cos(t) * cos((9 / 10 * t))), (2 * 10) + (sin(t) * cos((9 / 10 * t)))), ((2 * 10) + (cos(t) * cos((10 / 10 * t))), (2 * 10) + (sin(t) * cos((10 / 10 * t))))}}
Sequence(Text[k, ((2 * k), -0.5)], k, 1, 10) → Tw
Sequence(Text[k, (-0.5, (2 * k) - 0.3)], k, 1, 10) → Ts
CurveCartesian(30 + ((7 * cos(t)) * cos((a / b * t))), 12 + ((7 * sin(t)) * cos((a / b * t))), t, 0, (120 * pi)) → c

AI Drawing Prompt Reference

Draw a polar curve given by r = cos(a/b · t), where a and b are adjustable parameters controlling the frequency and overall shape. The curve is generated by sampling parameter t from 0 to 20π, forming a continuous smooth graph. The periodic structure and amplitude vary with the ratio a/b. Text labels can display the formula and parameter meanings to aid understanding.

The prompt above was compiled from the original drawing commands and is for reference only.