### Functions, Graphs, & Curves

• 1. 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 ∈ X relates to y ∈ Y , then we write y = f (x). X is called the domain and f (X) = {f (x) : x ∈ 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) = 2πt See Figure 1 Figure 1. Example of a Function 2. Curves in the Complex Plane A continuous function γ : [0, 1] → C is called a curve in the complex plane. The complex numbers γ(0), γ(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 γ(0) = γ(1). If γ(0) = γ(1) the curve is said to be open. 1
• 2. 2 SEBASTIAN VATTAMATTAM If f : X → Y is a function, then G(f ) = {(x, f (x)) : x ∈ X.} is the graph of f. The graph of the above function 1 is G(f ) = {(t, f (t)) : t ∈ [0, 1].} = {(t, 2πt) : t ∈ [0, 1].} For every value of t, (t, 2πt) is an ordered pair of real numbers, and hence it corresponds to the complex number t + i2πt. This gives the curve, representing the function  See Figure 2 (2.1) γ(t) = t + i2πt, t ∈ [0, 1] Figure 2. Graph of Function 1 3. Examples of Closed Curves 3.1. Unit Circle. (3.1) x2 + y 2 = 1 The parametric equations are x = cos(θ) y = sin(θ) As a curve, its equation is (3.2) u(t) = cos(2πt) + i sin(2πt) 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 ρ = cos(nθ), n ∈ N are called Rhodonea Cosine curves.
• 3. CURVES 3 Figure 3. Unit Circle 3.2.1. Rhodonea-Cosine:n = 1. (3.3) ρ = cos θ The parametric equations are x = cos(θ) cos(θ) y = cos(θ) sin(θ) As a curve, its equation is (3.4) ρ(t) = cos2 (2πt) + i cos(2πt) sin(2πt) ρ(1) = ρ(0) = 1 Therefore ρ is a loop at 1. See Figure 4 3.2.2. Rhodonea-Cosine:n = 2. (3.5) ρ = cos(2θ) The parametric equations are x = cos(2θ) cos(θ) y = cos(2θ) sin(θ) As a curve, its equation is (3.6) ρ(t) = cos(4πt) cos(2πt) + i cos(4πt) sin(2πt) ρ(1) = ρ(0) = 1 Therefore ρ is a loop at 1. See Figure 5
• 4. 4 SEBASTIAN VATTAMATTAM Figure 4. Rhodonea-Cosine: n = 1 Figure 5. Rhodonea Cosine: n = 2 3.2.3. Rhodonea-Cosine:n = 3. (3.7) ρ = cos(3θ) The parametric equations are x = cos(3θ) cos(θ) y = cos(3θ) sin(θ)
• 5. CURVES 5 As a curve, its equation is (3.8) ρ(t) = cos(6πt) cos(2πt) + i cos(6πt) sin(2πt) ρ(1) = ρ(0) = 1 Therefore ρ is a loop at 1. See Figure 6 Figure 6. Rhodonea-Cosine: n = 3 3.3. Rhodonea-Sine Curves. Curves with the polar equation ρ = sin(nθ), n ∈ N are called Rhodonea Sine curves. 3.3.1. Rhodonea-Sine: n = 1. (3.9) ρ = sin θ The parametric equations are x = sin(θ) cos(θ) y = sin(θ) sin(θ) As a curve, its equation is (3.10) ρ(t) = sin(2πt) cos(2πt) + i sin(2πt) sin(2πt) ρ(1) = ρ(0) = 0 Therefore ρ is a loop at 0. See Figure 6
• 6. 6 SEBASTIAN VATTAMATTAM Figure 7. Rhodonea-Sin: n = 1 3.3.2. Rhodonea-Sine: n = 2. (3.11) ρ = sin(2θ) The parametric equations are x = sin(2θ) cos(θ) y = sin(2θ) sin(θ) As a curve, its equation is (3.12) ρ(t) = sin(4πt) cos(2πt) + i sin(4πt) sin(2πt) ρ(1) = ρ(0) = 0 Therefore ρ is a loop at 0. See Figure 6 3.3.3. Rhodonea-Sine: n = 3. (3.13) ρ = sin(3θ) The parametric equations are x = sin(2θ) cos(θ) y = sin(2θ) sin(θ) As a curve, its equation is (3.14) ρ(t) = sin(6πt) cos(2πt) + i sin(6πt) sin(2πt) ρ(1) = ρ(0) = 0 Therefore ρ is a loop at 0. See Figure 9
• 7. CURVES 7 Figure 8. Rhodonea-Sin: n = 2 Figure 9. Rhodonea-Sin: n = 2 3.4. Cardioid. (3.15) ρ = 1 + cos θ The parametric equations are x = (1 + cos(θ)) cos(θ) y = (1 + cos(θ)) sin(θ)
• 8. 8 SEBASTIAN VATTAMATTAM As a curve, its equation is (3.16) γ(t) = (1 + cos(2πt)) cos(2πt) + i(1 + cos(2πt)) sin(2πt) γ(1) = 2, γ(0) = 2 Therefore ρ is a loop at 2. Now deﬁne ζ = γ − 1. Then ζ is a loop at 1 See Figure 10 Figure 10. Cardioid 3.5. Double Folium. (3.17) ρ = 4 cos(θ) sin(2θ) The parametric equations are x = 4 cos(θ) sin(2θ) cos(θ) y = 4 cos(θ) sin(2θ) sin(θ) As a curve, its equation is (3.18) γ(t) = 4 cos(2πt) sin(4πt) cos(2πt) + i4 cos(2πt) sin(4πt) sin(2πt) γ(1) = γ(0) = 0 Therefore γ is a loop at 0. Now deﬁne ϕ = γ + 1. Then ϕ is a loop at 1 See Figure 11 3.6. Limacon of Pascal. (3.19) ρ = 1 + 2 cos θ The parametric equations are x = (1 + 2 cos(θ)) cos(θ) y = (1 + 2 cos(θ)) sin(θ) As a curve, its equation is (3.20) γ(t) = (1 + 2 cos(2πt)) cos(2πt) + i(1 + 2 cos(2πt)) sin(2πt)
• 9. CURVES 9 Figure 11. Double Folium γ(1) = γ(0) = 3 Therefore ρ is a loop at 3. Now deﬁne λ = γ − 2. Then λ is a loop at 1 See Figure 12 Figure 12. Limacon of Pascal 3.7. Crooked Egg. (3.21) ρ = cos3 θ + sin3 θ
• 10. 10 SEBASTIAN VATTAMATTAM The parametric equations are x = (cos3 θ + sin3 θ) cos(θ) y = (cos3 θ + sin3 θ) sin(θ) As a curve, its equation is (3.22) γ(t) = (cos3 (2πt) + sin3 (2πt)) cos(2πt) + i(cos3 (2πt) + sin3 (2πt)) sin(2πt) γ(1) = γ(0) = 1 Therefore γ is a loop at 1. See Figure 13 Figure 13. Crooked Egg 3.8. Nephroid. x = −3 sin(2θ) + sin(6θ) y = 3 cos(2θ) − cos(6θ) (3.23) υ(t) = −3 sin(4πt) + sin(12πt) + i[3 cos(4πt) − cos(12πt)] υ(1) = υ(0) = 2i Therefore υ is a loop at 2i. So, γ = υ − 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 = t
• 11. CURVES 11 Figure 14. Nephroid As a curve its equation is (4.2) λ(t) = 1 − t + it λ(1) = i, λ(0) = 1 See Figure 15 Figure 15. A Line Segment
• 12. 12 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) π(t) = 2t − 1 + i(2t − 1)2 π(1) = 1 + i, π(0) = −1 + i See Figure 16 Figure 16. A Parabola 4.3. Cosine Curve. (4.5) y = cos θ, 0 ≤ θ ≤ 2π The parametric equations are x = 2πt y = cos(2πt) As a curve its equation is (4.6) γ(t) = 2πt + i cos(2πt) γ(1) = 2π + i, γ(0) = i See Figure ??
• 13. CURVES 13 Figure 17. Cosine Curve 4.4. Sine Curve. (4.7) y = sin θ, 0 ≤ θ ≤ 4π The parametric equations are x = 4πt y = sin(4πt) As a curve its equation is (4.8) γ(t) = 4πt + i sin(4πt) γ(1) = 4π, γ(0) = 0 See Figure ?? 4.5. Archimedean Spiral. (4.9) ρ = θ, 0 ≤ θ ≤ 4π The parametric equations are x = 4πt cos(4πt) y = 4πt sin(4πt) As a curve its equation is (4.10) σ(t) = 4πt cos(4πt) + i4πt sin(4πt) ρ(1) = 4π, ρ(0) = 0 Therefore ρ is an open curve. See Figure ??
• 14. 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 − theoretica lgebra.