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Homework Statement
Hello, as part of a computer science project, I have been tasked with writing an application that will generate an involute spur gear based on given criteria. To do this, I will need to be able to find the point of intersection between the involute curve (used to define the edge of each gear tooth) and a circle that can be at an arbitrary point. Below is my attempt at solving this problem. I was hoping someone might be able to give me a nudge in the correct direction, as I reach a point where I am unsure how to proceed. Any help would be appreciated, thanks in advance.
Homework Equations
Involute Curve Equation
x = A(cosT + TsinT)
y = A(sinT - TcosT)
Circle EquaTion
(x - J)^2 + (y - K)^2 = R^2
The Attempt at a Solution
Substitute involute curve equation into the circle equation and solve for T
((A(cosT + TsinT)) - J)^2 + ((A(sinT - TcosT)) - K)^2 = R^2
Expand...
(AcosT + ATsinT - J)^2 + (AsinT - ATcosT - K)^2 = R^2
(AcosT)^2 + AcosT ATsinT - J AcosT + AcosT ATsinT + (ATsinT)^2 - J ATsinT - J AcosT - J ATsinT + J^2 +
(AsinT)^2 - AsinT ATcosT - K AsinT - AsinT ATcosT + (ATcosT)^2 + K ATcosT - K AsinT + K ATcosT + K^2 = R^2
Grouping like terms...
(AcosT)^2 + 2(AcosT ATsinT) - 2(J AcosT) - 2(J ATsinT) + AcosT ATsinT + (ATsinT)^2 + J^2 +
(AsinT)^2 - 2(AsinT ATcosT) - 2(K AsinT) + 2(K ATcosT) - AsinT ATcosT + (ATcosT)^2 + K^2 = R^2
(RcosT RTsinT) and (-RsinT RTcosT) cancel each other out...
(AcosT)^2 - 2(J AcosT) - 2(J ATsinT) + AcosT ATsinT + (ATsinT)^2 + J^2 +
(AsinT)^2 - 2(K AsinT) - 2(K ATcosT) - AsinT ATcosT + (ATcosT)^2 + K^2 = R^2
cos^2 and sin^2 can be removed using Pythagorean identity...
cos^2T(A^2 + (AT)^2) - 2(J AcosT) - 2(J ATsinT) + AcosT ATsinT + J^2 +
sin^2T(A^2 + (AT)^2) - 2(K AsinT) - 2(K ATcosT) - AsinT ATcosT + K^2 = R^2
(A^2 + (AT)^2)(cos^2T + sin^2T) - 2(J AcosT) - 2(J ATsinT) + AcosT ATsinT + J^2 - 2(K AsinT) - 2(K ATcosT) - AsinT ATcosT + K^2 = R^2
(A^2 + (AT)^2)(1) - 2(J AcosT) - 2(J ATsinT) + AcosT ATsinT + J^2 - 2(K AsinT) - 2(K ATcosT) - AsinT ATcosT + K^2 = R^2
At this point, I am unsure how to proceed...
(A^2 + (AT)^2) - 2(J AcosT) - 2(J ATsinT) + AcosT ATsinT + J^2 - 2(K AsinT) - 2(K ATcosT) - AsinT ATcosT + K^2 = R^2
After struggling with this for quite a while, I can't seem to find anything to do at this point that seems to be making headway toward solving the equation for T...