Functions, Graphs, & Curves

BOOK OF BEAUTIFUL CURVES: CHAPTER 1 STANDARDCURVES

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 exactlyto one element in Y. If x ∈ X relates to y ∈ Y , then we write y = f(x). X is calledthe 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. Thecomplex 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) 6= γ(1)the curve is said to be open.

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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 henceit corresponds to the complex number t + i2πt. This gives the curve, representingthe function [1] 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 + y2 = 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.

CURVES 3

Figure 3. Unit Circle

3.2.1. Rhodonea-Cosine:n = 1.

(3.3) ρ = cos θ


x = cos(θ) cos(θ)y = cos(θ) sin(θ)


(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


(3.5) ρ = cos(2θ)


x = cos(2θ) cos(θ)y = cos(2θ) sin(θ)


(3.6) ρ(t) = cos(4πt) cos(2πt) + i cos(4πt) sin(2πt)



Figure 4. Rhodonea-Cosine: n = 1

Figure 5. Rhodonea Cosine: n = 2


(3.7) ρ = cos(3θ)


x = cos(3θ) cos(θ)y = cos(3θ) sin(θ)

CURVES 5


(3.8) ρ(t) = cos(6πt) cos(2πt) + i cos(6πt) sin(2πt)


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


x = sin(θ) cos(θ)y = sin(θ) sin(θ)


(3.10) ρ(t) = sin(2πt) cos(2πt) + i sin(2πt) sin(2πt)



Figure 7. Rhodonea-Sin: n = 1


(3.11) ρ = sin(2θ)


x = sin(2θ) cos(θ)y = sin(2θ) sin(θ)





(3.13) ρ = sin(3θ)


x = sin(2θ) cos(θ)y = sin(2θ) sin(θ)




CURVES 7



3.4. Cardioid.

(3.15) ρ = 1 + cos θ


x = (1 + cos(θ)) cos(θ)y = (1 + cos(θ)) sin(θ)



(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 aloop at 1 See Figure 10

Figure 10. Cardioid

3.5. Double Folium.

(3.17) ρ = 4 cos(θ) sin(2θ)


x = 4 cos(θ) sin(2θ) cos(θ)y = 4 cos(θ) sin(2θ) sin(θ)


(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 loopat 1 See Figure 11

3.6. Limacon of Pascal.

(3.19) ρ = 1 + 2 cos θ


x = (1 + 2 cos(θ)) cos(θ)y = (1 + 2 cos(θ)) sin(θ)


(3.20) γ(t) = (1 + 2 cos(2πt)) cos(2πt) + i(1 + 2 cos(2πt)) sin(2πt)

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 loopat 1 See Figure 12

Figure 12. Limacon of Pascal

3.7. Crooked Egg.

(3.21) ρ = cos3 θ + sin3 θ



x = (cos3 θ + sin3 θ) cos(θ)y = (cos3 θ + sin3 θ) sin(θ)


(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. SeeFigure 3.23

4. Examples of Open Curves

4.1. A Line Segment.

(4.1) x + y = 1, 0 ≤ x ≤ 1


x = 1− t

y = t

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


4.2. A Parabola.

(4.3) y = x2,−1 ≤ x ≤ 1


x = 2t− 1y = (2t− 1)2


(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


x = 2πt

y = cos(2πt)


(4.6) γ(t) = 2πt + i cos(2πt)

γ(1) = 2π + i, γ(0) = i See Figure ??

CURVES 13

Figure 17. Cosine Curve

4.4. Sine Curve.

(4.7) y = sin θ, 0 ≤ θ ≤ 4π


x = 4πt

y = sin(4πt)


(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


x = 4πt cos(4πt)y = 4πt sin(4πt)


(4.10) σ(t) = 4πt cos(4πt) + i4πt sin(4πt)

ρ(1) = 4π, ρ(0) = 0 Therefore ρ is an open curve. See Figure ??


Figure 18. Sine Curve

Figure 19. Archimedean Spiral

5. Conclusion

We have presented here only the fundamentals of a vast theory of curves. Theauthor has developed an algebraic system called Functional Theoretic Algebra anda method of transforming curves, called n-Curving. Those who are interested mayplease visit http : //en.wikipedia.org/wiki/Functional − theoreticalgebra.

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