Throwing light on certain concepts, which are found difficult by students, and even by teachers.

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 [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 + 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.