An Aesthetic Exploration of Relative Motions

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3. Case Study of Aesthetic Forms

3-2 Chaotic Multiple-level Relative Motions

Figure 3-2 Motions Illustration

Rules:

dAP₃ = Constant
(*dAP₃: dynamic angular increment of Point P₃)

P₀: P_0(x,y) = (0, 0)

P₁: P_1(angle) = P_1(angle) + P_{1(angular increment)}

P_1(x) = P_4(x)+ P_1(radius)* P_{1(cos(angle))}

P_1(y) = P_4(y)+ P_1(radius)* P_{1(sin(angle))}

P₂: P_2(angle) = P_2(angle) + P_{2(angular increment)}

P_2(x) = P_5(x)+ P_2(radius)* P_{2(cos(angle))}

P_2(y) = P_5(y)+ P_2(radius)* P_{2(sin(angle))}

P₃: P_3(angle) = P_3(angle) + P_{3(angular increment)}

P_3(x) = P_0(x)+ P_3(radius)* P_{3(cos(angle))}

P_3(y) = P_0(y)+ P_3(radius)* P_{3(sin(angle))}

P₄: P_4(x,y) = (P_2(x,y)+ P_3(x,y)) / 2

P₅: P_5(x,y) = (P_1(x,y)+ P_3(x,y)) / 2

P₆: P_6(x,y) = (P_1(x,y)+ P_2(x,y)+ P_3(x,y)) / 3

Trajectory Recorded: P₆

Table: Case 3-2.1 Parameters Assigned (Figure 3-2.1)

Point	P₁	P₂	P₃
Coordinate System	P₄	P₅	P₀
Initial Angle	0	0	0
Angular Increment	0.04	-0.36	ceiling limit: 0.42
Angular Increment	0.04	-0.36	floor limit: 0.24
Radius(pixel)	150	150	80
Radius Increment	None	None	None
Chaotic Terms	if P_3(y) is greater than or equal to P_6(y), then if P_{_{3(angular increment)}} >= dAP_{3(ceiling limit)} or P_{3(angular increment)}<= dAP_{3(floor limit)}, then set dAP₃ to dAP₃ * (-1) end if P_{3(angular increment)} = P_{3(angular increment)} + dAP₃ end if

[Enlarge the image]

Figure 3-2.1 (12,000,000 iterations)

dAP₃ = 0.00031
* Positive angle means clockwise rotation, negative counterclockwise.

Table: Case 3-2.2 Parameters Assigned (Figure 3-2.2)

Point	P₁	P₂	P₃
Coordinate System	P₄	P₅	P₀
Initial Angle	180	0	0
Angular Increment	0.32	0.24	ceiling limit: 0.42
Angular Increment	0.32	0.24	floor limit: -0.48
Radius(pixel)	150	150	80
Radius Increment	None	None	None
Chaotic Terms	if P_3(y) is greater than or equal to P_6(y), then if P_{_{3(angular increment)}} >= dAP_{3(ceiling limit)} or P_{3(angular increment)}<= dAP_{3(floor limit)}, then set dAP₃ to dAP₃ * (-1) end if P_{3(angular increment)} = P_{3(angular increment)} + dAP₃ else if P_3(y) is less than P_6(y), then set P_{3(angular increment)} to P_{3(angular increment)} * (-1) end if

[Enlarge the image]

Figure 3-2.2 (12,000,000 iterations)

dAP₃ = 0.003
* Positive angle means clockwise rotation, negative counterclockwise.

Table: Case 3-2.3 Parameters Assigned (Figure 3-2.3)

Point	P₁	P₂	P₃
Coordinate System	P₄	P₅	P₀
Initial Angle	-180	0	0
Angular Increment	0.32	0.24	ceiling limit: 0.82
Angular Increment	0.32	0.24	floor limit: -0.48
Radius(pixel)	150	100	80
Radius Increment	None	None	None
Chaotic Terms	if P_3(y) is greater than or equal to P_6(y), then if P_{_{3(angular increment)}} >= dAP_{3(ceiling limit)} or P_{3(angular increment)}<= dAP_{3(floor limit)}, then set dAP₃ to dAP₃ * (-1) end if P_{3(angular increment)} = P_{3(angular increment)} + dAP₃ else if P_3(y) is less than P_6(y), then set P_{3(angular increment)} to P_{3(angular increment)} * (-1) end if

[Enlarge the image]

Figure 3-2.3 (12,000,000 iterations)

dAP₃ = 0.007
* Positive angle means clockwise rotation, negative counterclockwise.

Table: Case 3-2.4 Parameters Assigned (Figure 3-2.4)

Point	P₁	P₂	P₃
Coordinate System	P₄	P₅	P₀
Initial Angle	0	0	0
Angular Increment	-0.16	0.24	ceiling limit: 0.560036
Angular Increment	-0.16	0.24	floor limit: -0.480700
Radius(pixel)	240	120	60
Radius Increment	None	None	None
Chaotic Terms	if P_3(y) is greater than or equal to P_6(y), then set P_{3(angular increment)} to P_{3(angular increment)} * (-1) else if P_3(y) is less than P_6(y), then if P_{_{3(angular increment)}} >= dAP_{3(ceiling limit)} or P_{3(angular increment)}<= dAP_{3(floor limit)}, then set dAP₃ to dAP₃ * (-1) end if P_{3(angular increment)} = P_{3(angular increment)} + dAP₃ end if

[Enlarge the image]

Figure 3-2.4 (12,000,000 iterations)

dAP₃ = 0.023
* Positive angle means clockwise rotation, negative counterclockwise.

Table: Case 3-2.5 Parameters Assigned (Figure 3-2.5)

Point	P₁	P₂	P₃
Coordinate System	P₄	P₅	P₀
Initial Angle	0	0	0
Angular Increment	0.2	-0.24	ceiling limit: 0.160036
Angular Increment	0.2	-0.24	floor limit: -0.480700
Radius(pixel)	240	120	60
Radius Increment	None	None	None
Chaotic Terms	if P_3(y) is greater than or equal to P_6(y), then set P_{3(angular increment)} to P_{3(angular increment)} * (-1) else if P_3(y) is less than P_6(y), then if P_{_{3(angular increment)}} >= dAP_{3(ceiling limit)} or P_{3(angular increment)}<= dAP_{3(floor limit)}, then set dAP₃ to dAP₃ * (-1) end if P_{3(angular increment)} = P_{3(angular increment)} + dAP₃ end if

[Enlarge the image]

Figure 3-2.5 (12,000,000 iterations)

dAP₃ = 0.004
* Positive angle means clockwise rotation, negative counterclockwise.

Table: Case 3-2.6 Parameters Assigned (Figure 3-2.6)

Point	P₁	P₂	P₃
Coordinate System	P₄	P₅	P₀
Initial Angle	0	0	0
Angular Increment	0.18	-0.24	ceiling limit: 0.240036
Angular Increment	0.18	-0.24	floor limit: -0.480700
Radius(pixel)	240	120	60
Radius Increment	None	None	None
Chaotic Terms	if P_3(y) is greater than or equal to P_6(y), then set P_{3(angular increment)} to P_{3(angular increment)} * (-1) else if P_3(y) is less than P_6(y), then if P_{_{3(angular increment)}} >= dAP_{3(ceiling limit)} or P_{3(angular increment)}<= dAP_{3(floor limit)}, then set dAP₃ to dAP₃ * (-1) end if P_{3(angular increment)} = P_{3(angular increment)} + dAP₃ end if

[Enlarge the image]

Figure 3-2.6 (12,000,000 iterations)

dAP₃ = 0.004
* Positive angle means clockwise rotation, negative counterclockwise.

Table: Case 3-2.7 Parameters Assigned (Figure 3-2.7)

Point	P₁	P₂	P₃
Coordinate System	P₄	P₅	P₀
Initial Angle	0	0	0
Angular Increment	-0.24	ceiling limit: 0.18	ceiling limit: 0.48
Angular Increment	-0.24	floor limit: 0.12	floor limit: 0.26
Radius(pixel)	150	150	80
Radius Increment	None	None	None
Chaotic Terms	P₂: if P_{_{2(angular increment)}} >= dAP_{2(ceiling limit)} or P_{2(angular increment)}<= dAP_{2(floor limit)}, then set dAP₂ to dAP₂ * (-1) end if P₃: if P_3(y) is less than P_6(y), then if P_{_{3(angular increment)}} >= dAP_{3(ceiling limit)} or P_{3(angular increment)}<= dAP_{3(floor limit)}, then set dAP₃ to dAP₃ * (-1) end if P_{3(angular increment)} = P_{3(angular increment)} + dAP₃ end if

[Enlarge the image]

Figure 3-2.7 (12,000,000 iterations)

dAP₂ = -0.003
dAP₃ = -0.008
* Positive angle means clockwise rotation, negative counterclockwise.

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CONTENTS:

Index

Introduction

Methodology

Case Study of Aesthetic Forms

Conclusions