z-Plane <-> w-Plane

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2022-06-11 01:01

This example demonstrates the transformation of the complex function w = z + 1/z. Key mathematical concepts include complex numbers, transformations, and coordinate geometry. Using GeoGebra commands such as CurveCartesian, Point, and Textfield, the z-plane is mapped to the w-plane. Suitable for high school mathematics education and learning complex number transformations.

GeoGebra Commands & Parameters

CurveCartesian(X(t), Y(t), t, a, b) → zkurve
Point(zkurve) → A
Textfield(X) → Eingabefeld1
Textfield(Y) → Eingabefeld2
Textfield(a) → Eingabefeld3
Textfield(b) → Eingabefeld4
Sequence(CurveCartesian[(r * X(t)), (r * Y(t)), t, a, b], r, 0, 10, 0.5) → zKL, {((0 * cos(t) / 2), (0 * sin(t))), ((0.5 * cos(t) / 2), (0.5 * sin(t))), ((1 * cos(t) / 2), (1 * sin(t))), ((1.5 * cos(t) / 2), (1.5 * sin(t))), ((2 * cos(t) / 2), (2 * sin(t))), ((2.5 * cos(t) / 2), (2.5 * sin(t))), ((3 * cos(t) / 2), (3 * sin(t))), ((3.5 * cos(t) / 2), (3.5 * sin(t))), ((4 * cos(t) / 2), (4 * sin(t))), ((4.5 * cos(t) / 2), (4.5 * sin(t))), ((5 * cos(t) / 2), (5 * sin(t))), ((5.5 * cos(t) / 2), (5.5 * sin(t))), ((6 * cos(t) / 2), (6 * sin(t))), ((6.5 * cos(t) / 2), (6.5 * sin(t))), ((7 * cos(t) / 2), (7 * sin(t))), ((7.5 * cos(t) / 2), (7.5 * sin(t))), ((8 * cos(t) / 2), (8 * sin(t))), ((8.5 * cos(t) / 2), (8.5 * sin(t))), ((9 * cos(t) / 2), (9 * sin(t))), ((9.5 * cos(t) / 2), (9.5 * sin(t))), ((10 * cos(t) / 2), (10 * sin(t)))}
Sequence(CurveCartesian[(r * X(t)) + X(t) / ((r * (X(t))^(2)) + (r * (Y(t))^(2))), (r * Y(t)) - Y(t) / ((r * (X(t))^(2)) + (r * (Y(t))^(2))), t, a, b], r, 0, 10, 0.5) → wKL, {((0 * cos(t) / 2) + cos(t) / 2 / ((0 * (cos(t) / 2)^(2)) + (0 * (sin(t))^(2))), (0 * sin(t)) - sin(t) / ((0 * (cos(t) / 2)^(2)) + (0 * (sin(t))^(2)))), ((0.5 * cos(t) / 2) + cos(t) / 2 / ((0.5 * (cos(t) / 2)^(2)) + (0.5 * (sin(t))^(2))), (0.5 * sin(t)) - sin(t) / ((0.5 * (cos(t) / 2)^(2)) + (0.5 * (sin(t))^(2)))), ((1 * cos(t) / 2) + cos(t) / 2 / ((1 * (cos(t) / 2)^(2)) + (1 * (sin(t))^(2))), (1 * sin(t)) - sin(t) / ((1 * (cos(t) / 2)^(2)) + (1 * (sin(t))^(2)))), ((1.5 * cos(t) / 2) + cos(t) / 2 / ((1.5 * (cos(t) / 2)^(2)) + (1.5 * (sin(t))^(2))), (1.5 * sin(t)) - sin(t) / ((1.5 * (cos(t) / 2)^(2)) + (1.5 * (sin(t))^(2)))), ((2 * cos(t) / 2) + cos(t) / 2 / ((2 * (cos(t) / 2)^(2)) + (2 * (sin(t))^(2))), (2 * sin(t)) - sin(t) / ((2 * (cos(t) / 2)^(2)) + (2 * (sin(t))^(2)))), ((2.5 * cos(t) / 2) + cos(t) / 2 / ((2.5 * (cos(t) / 2)^(2)) + (2.5 * (sin(t))^(2))), (2.5 * sin(t)) - sin(t) / ((2.5 * (cos(t) / 2)^(2)) + (2.5 * (sin(t))^(2)))), ((3 * cos(t) / 2) + cos(t) / 2 / ((3 * (cos(t) / 2)^(2)) + (3 * (sin(t))^(2))), (3 * sin(t)) - sin(t) / ((3 * (cos(t) / 2)^(2)) + (3 * (sin(t))^(2)))), ((3.5 * cos(t) / 2) + cos(t) / 2 / ((3.5 * (cos(t) / 2)^(2)) + (3.5 * (sin(t))^(2))), (3.5 * sin(t)) - sin(t) / ((3.5 * (cos(t) / 2)^(2)) + (3.5 * (sin(t))^(2)))), ((4 * cos(t) / 2) + cos(t) / 2 / ((4 * (cos(t) / 2)^(2)) + (4 * (sin(t))^(2))), (4 * sin(t)) - sin(t) / ((4 * (cos(t) / 2)^(2)) + (4 * (sin(t))^(2)))), ((4.5 * cos(t) / 2) + cos(t) / 2 / ((4.5 * (cos(t) / 2)^(2)) + (4.5 * (sin(t))^(2))), (4.5 * sin(t)) - sin(t) / ((4.5 * (cos(t) / 2)^(2)) + (4.5 * (sin(t))^(2)))), ((5 * cos(t) / 2) + cos(t) / 2 / ((5 * (cos(t) / 2)^(2)) + (5 * (sin(t))^(2))), (5 * sin(t)) - sin(t) / ((5 * (cos(t) / 2)^(2)) + (5 * (sin(t))^(2)))), ((5.5 * cos(t) / 2) + cos(t) / 2 / ((5.5 * (cos(t) / 2)^(2)) + (5.5 * (sin(t))^(2))), (5.5 * sin(t)) - sin(t) / ((5.5 * (cos(t) / 2)^(2)) + (5.5 * (sin(t))^(2)))), ((6 * cos(t) / 2) + cos(t) / 2 / ((6 * (cos(t) / 2)^(2)) + (6 * (sin(t))^(2))), (6 * sin(t)) - sin(t) / ((6 * (cos(t) / 2)^(2)) + (6 * (sin(t))^(2)))), ((6.5 * cos(t) / 2) + cos(t) / 2 / ((6.5 * (cos(t) / 2)^(2)) + (6.5 * (sin(t))^(2))), (6.5 * sin(t)) - sin(t) / ((6.5 * (cos(t) / 2)^(2)) + (6.5 * (sin(t))^(2)))), ((7 * cos(t) / 2) + cos(t) / 2 / ((7 * (cos(t) / 2)^(2)) + (7 * (sin(t))^(2))), (7 * sin(t)) - sin(t) / ((7 * (cos(t) / 2)^(2)) + (7 * (sin(t))^(2)))), ((7.5 * cos(t) / 2) + cos(t) / 2 / ((7.5 * (cos(t) / 2)^(2)) + (7.5 * (sin(t))^(2))), (7.5 * sin(t)) - sin(t) / ((7.5 * (cos(t) / 2)^(2)) + (7.5 * (sin(t))^(2)))), ((8 * cos(t) / 2) + cos(t) / 2 / ((8 * (cos(t) / 2)^(2)) + (8 * (sin(t))^(2))), (8 * sin(t)) - sin(t) / ((8 * (cos(t) / 2)^(2)) + (8 * (sin(t))^(2)))), ((8.5 * cos(t) / 2) + cos(t) / 2 / ((8.5 * (cos(t) / 2)^(2)) + (8.5 * (sin(t))^(2))), (8.5 * sin(t)) - sin(t) / ((8.5 * (cos(t) / 2)^(2)) + (8.5 * (sin(t))^(2)))), ((9 * cos(t) / 2) + cos(t) / 2 / ((9 * (cos(t) / 2)^(2)) + (9 * (sin(t))^(2))), (9 * sin(t)) - sin(t) / ((9 * (cos(t) / 2)^(2)) + (9 * (sin(t))^(2)))), ((9.5 * cos(t) / 2) + cos(t) / 2 / ((9.5 * (cos(t) / 2)^(2)) + (9.5 * (sin(t))^(2))), (9.5 * sin(t)) - sin(t) / ((9.5 * (cos(t) / 2)^(2)) + (9.5 * (sin(t))^(2)))), ((10 * cos(t) / 2) + cos(t) / 2 / ((10 * (cos(t) / 2)^(2)) + (10 * (sin(t))^(2))), (10 * sin(t)) - sin(t) / ((10 * (cos(t) / 2)^(2)) + (10 * (sin(t))^(2))))}
CurveCartesian(U(t), V(t), t, a, b) → wkurve

AI Drawing Prompt Reference

On a coordinate plane, choose two adjustable parameters a and b, and define a parameter t running from a to b. Draw a z-plane curve by taking its x- and y-coordinates as X(t) and Y(t), so each point corresponds to z = X(t) + iY(t). Then map every z point to the w-plane using the complex transformation w = z + 1/z. Compute the real part and imaginary part of w from this formula and use them as the x- and y-coordinates, respectively, to generate the corresponding w-plane curve with the same parameter t range. Show the resulting w-curve together with the original z-curve, with a and b adjustable through input fields.

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