BOOK OF BEAUTIFUL CURVES: CHAPTER 1 STANDARD CURVES SEBASTIAN VATTAMATTAM 1. What is a Function ? If X, Y are two sets, a relation f from X to Y is a function if f relates exactly to one element in Y. If x 2 X relates to y 2 Y , then we write y = f(x). X is called the domain and f(X) = {f(x) : x 2 X} is called the range of f. Example 1.1. X = [0, 1], Y = R. (1.1) f : [0, 1] ! R, defined by f(t) = 2t See Figure 1 Figure 1. Example of a Function 2. Curves in the Complex Plane A continuous function : [0, 1] ! C is called a curve in the complex plane. The complex numbers (0), (1) are called the end points of the curve. If they are equal, the curve is said to be closed, and we call it a loop at (0) = (1). If (0) 6= (1) the curve is said to be open. 12 SEBASTIAN VATTAMATTAM If f : X ! Y is a function, then G(f) = {(x, f(x)) : x 2 X.} is the graph of f. The graph of the above function 1 is G(f) = {(t, f(t)) : t 2 [0, 1].} = {(t, 2t) : t 2 [0, 1].} For every value of t, (t, 2t) is an ordered pair of real numbers, and hence it corresponds to the complex number t + i2t. This gives the curve, representing the function [1] See Figure 2 (2.1) (t) = t + i2t, t 2 [0, 1] Figure 2. Graph of Function 1 3. Examples of Closed Curves 3.1. Unit Circle. (3.1) x2 + y2 = 1 The parametric equations are x = cos() y = sin() As a curve, its equation is (3.2) u(t) = cos(2t) + i sin(2t) u(1) = u(0) = 1 Therefore u is a loop at 1. See Figure 3 3.2. Rhodonea-Cosine Curves. Curves with the polar equation = cos(n), n 2 N are called Rhodonea Cosine curves.CURVES 3 Figure 3. Unit Circle 3.2.1. Rhodonea-Cosine:n = 1. (3.3) = cos The parametric equations are x = cos() cos() y = cos() sin() As a curve, its equation is (3.4) (t) = cos2(2t) + i cos(2t) sin(2t) (1) = (0) = 1 Therefore is a loop at 1. See Figure 4 3.2.2. Rhodonea-Cosine:n = 2. (3.5) = cos(2) The parametric equations are x = cos(2) cos() y = cos(2) sin() As a curve, its equation is (3.6) (t) = cos(4t) cos(2t) + i cos(4t) sin(2t) (1) = (0) = 1 Therefore is a loop at 1. See Figure 54 SEBASTIAN VATTAMATTAM Figure 4. Rhodonea-Cosine: n = 1 Figure 5. Rhodonea Cosine: n = 2 3.2.3. Rhodonea-Cosine:n = 3. (3.7) = cos(3) The parametric equations are x = cos(3) cos() y = cos(3) sin()CURVES 5 As a curve, its equation is (3.8) (t) = cos(6t) cos(2t) + i cos(6t) sin(2t) (1) = (0) = 1 Therefore is a loop at 1. See Figure 6 Figure 6. Rhodonea-Cosine: n = 3 3.3. Rhodonea-Sine Curves. Curves with the polar equation = sin(n), n 2 N are called Rhodonea Sine curves. 3.3.1. Rhodonea-Sine: n = 1. (3.9) = sin The parametric equations are x = sin() cos() y = sin() sin() As a curve, its equation is (3.10) (t) = sin(2t) cos(2t) + i sin(2t) sin(2t) (1) = (0) = 0 Therefore is a loop at 0. See Figure 66 SEBASTIAN VATTAMATTAM Figure 7. Rhodonea-Sin: n = 1 3.3.2. Rhodonea-Sine: n = 2. (3.11) = sin(2) The parametric equations are x = sin(2) cos() y = sin(2) sin() As a curve, its equation is (3.12) (t) = sin(4t) cos(2t) + i sin(4t) sin(2t) (1) = (0) = 0 Therefore is a loop at 0. See Figure 6 3.3.3. Rhodonea-Sine: n = 3. (3.13) = sin(3) The parametric equations are x = sin(2) cos() y = sin(2) sin() As a curve, its equation is (3.14) (t) = sin(6t) cos(2t) + i sin(6t) sin(2t) (1) = (0) = 0 Therefore is a loop at 0. See Figure 9CURVES 7 Figure 8. Rhodonea-Sin: n = 2 Figure 9. Rhodonea-Sin: n = 2 3.4. Cardioid. (3.15) = 1 + cos The parametric equations arex = (1 + cos()) cos() y = (1 + cos()) sin()8 SEBASTIAN VATTAMATTAM As a curve, its equation is (3.16) (t) = (1 + cos(2t)) cos(2t) + i(1 + cos(2t)) sin(2t) (1) = 2, (0) = 2 Therefore is a loop at 2. Now define = − 1. Then is a loop at 1 See Figure 10 Figure 10. Cardioid 3.5. Double Folium. (3.17) = 4 cos() sin(2) The parametric equations are x = 4 cos() sin(2) cos() y = 4 cos() sin(2) sin() As a curve, its equation is (3.18) (t) = 4 cos(2t) sin(4t) cos(2t) + i4 cos(2t) sin(4t) sin(2t) (1) = (0) = 0 Therefore is a loop at 0. Now define ' = +1. Then ' is a loop at 1 See Figure 11 3.6. Limacon of Pascal. (3.19) = 1 + 2 cos The parametric equations are x = (1 + 2 cos()) cos() y = (1 + 2 cos()) sin() As a curve, its equation is (3.20) (t) = (1 + 2 cos(2t)) cos(2t) + i(1 + 2 cos(2t)) sin(2t)CURVES 9 Figure 11. Double Folium (1) = (0) = 3 Therefore is a loop at 3. Now define = −2. Then is a loop at 1 See Figure 12 Figure 12. Limacon of Pascal 3.7. Crooked Egg. (3.21) = cos3 + sin3 10 SEBASTIAN VATTAMATTAM The parametric equations are x = (cos3 + sin3 ) cos() y = (cos3 + sin3 ) sin() As a curve, its equation is (3.22) (t) = (cos3(2t) + sin3(2t)) cos(2t) + i(cos3(2t) + sin3(2t)) sin(2t) (1) = (0) = 1 Therefore is a loop at 1. See Figure 13 Figure 13. Crooked Egg 3.8. Nephroid. x = −3 sin(2) + sin(6) y = 3 cos(2) − cos(6) (3.23) (t) = −3 sin(4t) + sin(12t) + i[3 cos(4t) − cos(12t)] (1) = (0) = 2i Therefore is a loop at 2i. So, = − 2i is a loop at 1. See Figure 3.23 4. Examples of Open Curves 4.1. A Line Segment. (4.1) x + y = 1, 0 x 1 The parametric equations are x = 1 − t y = tCURVES 11 Figure 14. Nephroid As a curve its equation is (4.2) (t) = 1 − t + it (1) = i, (0) = 1 See Figure 15 Figure 15. A Line Segment12 SEBASTIAN VATTAMATTAM 4.2. A Parabola. (4.3) y = x2,−1 x 1 The parametric equations are x = 2t − 1 y = (2t − 1)2 As a curve its equation is (4.4) (t) = 2t − 1 + i(2t − 1)2 (1) = 1 + i, (0) = −1 + i See Figure 16 Figure 16. A Parabola 4.3. Cosine Curve. (4.5) y = cos , 0 2The parametric equations are x = 2t y = cos(2t) As a curve its equation is (4.6) (t) = 2t + i cos(2t) (1) = 2+ i, (0) = i See Figure ??CURVES 13 Figure 17. Cosine Curve 4.4. Sine Curve. (4.7) y = sin , 0 4The parametric equations are x = 4t y = sin(4t) As a curve its equation is (4.8) (t) = 4t + i sin(4t) (1) = 4, (0) = 0 See Figure ?? 4.5. Archimedean Spiral. (4.9) = , 0 4The parametric equations are x = 4t cos(4t) y = 4t sin(4t) As a curve its equation is (4.10) (t) = 4t cos(4t) + i4t sin(4t) (1) = 4, (0) = 0 Therefore is an open curve. See Figure ??14 SEBASTIAN VATTAMATTAM Figure 18. Sine Curve Figure 19. Archimedean Spiral 5. Conclusion We have presented here only the fundamentals of a vast theory of curves. The author has developed an algebraic system called Functional Theoretic Algebra and a method of transforming curves, called n-Curving. Those who are interested may please visit http : //en.wikipedia.org/wiki/F unctional − theoreticalgebra.