Functions, Graphs, & Curves
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Transcript of 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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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
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
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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)
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
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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
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)
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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 aloop 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 loopat 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)
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
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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. SeeFigure 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
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
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4.2. A Parabola.
(4.3) y = x2,−1 ≤ x ≤ 1
The parametric equations are
x = 2t− 1y = (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 ??
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 ??
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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.