Particle Path Equation
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Particle Path Equation

Find the equation of path pattern of a particle.
The particle coordinate (x,y) is given by the functions
x = 2Sin(t) and y = 3Cos(t)


HOW TO USE: Enter new number on white input boxes. Answer is automatic. This template plus your smartphone will help you to calculate solution for similar problem in 60 seconds. To prove it, what is the equation of the path pattern of a particle if y = 7 Cos(t)
Answer: 49 x2 + 4 y2 = 196


x = Sin (t)

y = Cos (t)

SOLUTION

Sin(t) = ( x / ) equation 1
Cos (t) = ( y / ) equation 2

Formula recall
Sin2(t) + Cos2(t) = 1 , Trigonometric identity

By substitution: Cos2(t) = ( y / 3 )2 and Sin2(t) = ( x / 2 )2
( x2 / ) + ( y 2/ ) = 1

ANSWER, x2 + y2 = This is the equation of the path pattern of a particle.

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If you enter in Geogebra input box shown below the discovered path pattern equation of the particle. You will see the particle is traveling in elliptical pattern.

Type in Geogebra the following input: 9x^2 + 4y^2 = 36 then press the enter button in your laptop.



If you overlay the discovered equation in your scattered plot you are doing clustering. Clustering is unsupervised machine learning. I believe using human expert system like discovering the generalized equation and then overlaying it on top of the scattered plot is much faster approach, the prediction model is explainable, easy to deploy and almost 100% accurate and repeatable as long as the two principal component input independent variable " x " and " y " are within the specified boundary limits of each cluster equation.

Graphing Ellipse Using Geogebra

Generating Ellipse From Sine Wave Animation




By Apolinario "Sam" Ortega, 21 September 2018, Created with GeoGebra

Generating Sine Wave from Circle Animation

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