Exploring Conformal Mapping

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2022-01-25 02:52

This example demonstrates the concept of conformal mapping, which is a mapping that preserves angles in the complex plane. By adjusting parameters, observe the effect of different functions under conformal mapping. Commands used include Point, Sequence, Locus, Textfield, Polygon, Surface, etc. Suitable for complex function studies in university mathematics courses.

GeoGebra Commands & Parameters

Point(yAxis) → C
Point(xAxis) → A
Sequence(Segment[A + (0, y(C) + (k * dy)), ((-x(A)), y(C) + (k * dy))], k, 0, k0) → l1, {7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807, 7.42658575340807}
Point(g1) → z_1
Point(l1) → E
Locus(fz_{1}, z_1) → loc1
Sequence(CurveCartesian[real(f(u, v)), imaginary(f(u, v)), u, -10, 10], v, -10, 10) → l_{uv}, {(real(1 / (z + (0 * ί))(u, -10)), imaginary(1 / (z + (0 * ί))(u, -10))), (real(1 / (z + (0 * ί))(u, -9)), imaginary(1 / (z + (0 * ί))(u, -9))), (real(1 / (z + (0 * ί))(u, -8)), imaginary(1 / (z + (0 * ί))(u, -8))), (real(1 / (z + (0 * ί))(u, -7)), imaginary(1 / (z + (0 * ί))(u, -7))), (real(1 / (z + (0 * ί))(u, -6)), imaginary(1 / (z + (0 * ί))(u, -6))), (real(1 / (z + (0 * ί))(u, -5)), imaginary(1 / (z + (0 * ί))(u, -5))), (real(1 / (z + (0 * ί))(u, -4)), imaginary(1 / (z + (0 * ί))(u, -4))), (real(1 / (z + (0 * ί))(u, -3)), imaginary(1 / (z + (0 * ί))(u, -3))), (real(1 / (z + (0 * ί))(u, -2)), imaginary(1 / (z + (0 * ί))(u, -2))), (real(1 / (z + (0 * ί))(u, -1)), imaginary(1 / (z + (0 * ί))(u, -1))), (real(1 / (z + (0 * ί))(u, 0)), imaginary(1 / (z + (0 * ί))(u, 0))), (real(1 / (z + (0 * ί))(u, 1)), imaginary(1 / (z + (0 * ί))(u, 1))), (real(1 / (z + (0 * ί))(u, 2)), imaginary(1 / (z + (0 * ί))(u, 2))), (real(1 / (z + (0 * ί))(u, 3)), imaginary(1 / (z + (0 * ί))(u, 3))), (real(1 / (z + (0 * ί))(u, 4)), imaginary(1 / (z + (0 * ί))(u, 4))), (real(1 / (z + (0 * ί))(u, 5)), imaginary(1 / (z + (0 * ί))(u, 5))), (real(1 / (z + (0 * ί))(u, 6)), imaginary(1 / (z + (0 * ί))(u, 6))), (real(1 / (z + (0 * ί))(u, 7)), imaginary(1 / (z + (0 * ί))(u, 7))), (real(1 / (z + (0 * ί))(u, 8)), imaginary(1 / (z + (0 * ί))(u, 8))), (real(1 / (z + (0 * ί))(u, 9)), imaginary(1 / (z + (0 * ί))(u, 9))), (real(1 / (z + (0 * ί))(u, 10)), imaginary(1 / (z + (0 * ί))(u, 10)))}
Sequence(CurveCartesian[real(f(u, v)), imaginary(f(u, v)), v, -10, 10], u, -10, 10) → l_{vu}, {(real(1 / (z + (0 * ί))(-10, v)), imaginary(1 / (z + (0 * ί))(-10, v))), (real(1 / (z + (0 * ί))(-9, v)), imaginary(1 / (z + (0 * ί))(-9, v))), (real(1 / (z + (0 * ί))(-8, v)), imaginary(1 / (z + (0 * ί))(-8, v))), (real(1 / (z + (0 * ί))(-7, v)), imaginary(1 / (z + (0 * ί))(-7, v))), (real(1 / (z + (0 * ί))(-6, v)), imaginary(1 / (z + (0 * ί))(-6, v))), (real(1 / (z + (0 * ί))(-5, v)), imaginary(1 / (z + (0 * ί))(-5, v))), (real(1 / (z + (0 * ί))(-4, v)), imaginary(1 / (z + (0 * ί))(-4, v))), (real(1 / (z + (0 * ί))(-3, v)), imaginary(1 / (z + (0 * ί))(-3, v))), (real(1 / (z + (0 * ί))(-2, v)), imaginary(1 / (z + (0 * ί))(-2, v))), (real(1 / (z + (0 * ί))(-1, v)), imaginary(1 / (z + (0 * ί))(-1, v))), (real(1 / (z + (0 * ί))(0, v)), imaginary(1 / (z + (0 * ί))(0, v))), (real(1 / (z + (0 * ί))(1, v)), imaginary(1 / (z + (0 * ί))(1, v))), (real(1 / (z + (0 * ί))(2, v)), imaginary(1 / (z + (0 * ί))(2, v))), (real(1 / (z + (0 * ί))(3, v)), imaginary(1 / (z + (0 * ί))(3, v))), (real(1 / (z + (0 * ί))(4, v)), imaginary(1 / (z + (0 * ί))(4, v))), (real(1 / (z + (0 * ί))(5, v)), imaginary(1 / (z + (0 * ί))(5, v))), (real(1 / (z + (0 * ί))(6, v)), imaginary(1 / (z + (0 * ί))(6, v))), (real(1 / (z + (0 * ί))(7, v)), imaginary(1 / (z + (0 * ί))(7, v))), (real(1 / (z + (0 * ί))(8, v)), imaginary(1 / (z + (0 * ί))(8, v))), (real(1 / (z + (0 * ί))(9, v)), imaginary(1 / (z + (0 * ί))(9, v))), (real(1 / (z + (0 * ί))(10, v)), imaginary(1 / (z + (0 * ί))(10, v)))}
Sequence(Segment[C + (x(A) + (k * dx), 0), (0, (-y(C))) + (x(A) + (k * dx), 0)], k, 0, k0) → l2, {5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664, 5.623720789100664}
Point(g2) → z_{2}
Locus(fz_{2}, z_{2}) → loc2
Textfield(k0) → InputBox1
Textfield(m1) → InputBox2
Textfield(m2) → InputBox3
Polygon(D, F, n) → poli1, s, l, J, N, R, V, H, I, K, L, t, f_1, j_1, n_1, s_1, i, j, k
Point(poli1) → z_3
Locus(fz_{3}, z_3) → loc3
Surface(f(u + (v * ί)), u, -2, 2, v, -2, 2) → p

AI Drawing Prompt Reference

Draw a graph demonstrating the concept of conformal mapping in the complex plane. Start by plotting points on the x-axis and y-axis, then generate a series of line segments based on these points to observe their distribution in the complex plane. Next, draw the loci of these points under a complex function mapping to show the properties of conformal mapping. Finally, plot a surface representing a complex function to visually illustrate the effect of conformal mapping.

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