Expression 10: alpha left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis equals negative arc tangent left parenthesis, "f" prime left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis , right parenthesisαx0=−arctanf′x0
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elapsed angle since x=0
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Expression 12: theta Subscript, 1 , Baseline left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis equals StartFraction, 1 Over "r" , EndFraction Start integral from 0 to "x" Subscript, 0 , Baseline , end integral, StartRoot, 1 plus "f" prime left parenthesis, "t" , right parenthesis squared , EndRoot "d" "t"θ1x0=1r∫x001+f′t2dt
12
The equation for the circle and the center
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Expression 14: left parenthesis, "x" minus left parenthesis, "x" Subscript, 0 , Baseline plus "r" sin left parenthesis, alpha left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right parenthesis , right parenthesis squared plus left parenthesis, "y" minus left parenthesis, "f" left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis plus "r" cos left parenthesis, alpha left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right parenthesis , right parenthesis squared equals "r" squaredx−x0+rsinαx02+y−fx0+rcosαx02=r2
14
Expression 15: left parenthesis, "x" Subscript, 0 , Baseline plus "r" sin left parenthesis, alpha left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis , right parenthesis , "f" left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis plus "r" cos left parenthesis, alpha left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right parenthesisx0+rsinαx0,fx0+rcosαx0
15
Roulette point and equation
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Expression 17: left parenthesis, "x" Subscript, 0 , Baseline plus "r" times left parenthesis, sin left parenthesis, alpha left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis , right parenthesis minus sin left parenthesis, alpha left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis plus theta Subscript, 1 , Baseline left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right parenthesis , "f" left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis plus "r" times left parenthesis, cos left parenthesis, alpha left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis , right parenthesis minus cos left parenthesis, alpha left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis plus theta Subscript, 1 , Baseline left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right parenthesis , right parenthesisx0+r·sinαx0−sinαx0+θ1x0,fx0+r·cosαx0−cosαx0+θ1x0
17
Expression 18: left parenthesis, "t" plus "r" times left parenthesis, sin left parenthesis, alpha left parenthesis, "t" , right parenthesis , right parenthesis minus sin left parenthesis, alpha left parenthesis, "t" , right parenthesis plus theta Subscript, 1 , Baseline left parenthesis, "t" , right parenthesis , right parenthesis , right parenthesis , "f" left parenthesis, "t" , right parenthesis plus "r" times left parenthesis, cos left parenthesis, alpha left parenthesis, "t" , right parenthesis , right parenthesis minus cos left parenthesis, alpha left parenthesis, "t" , right parenthesis plus theta Subscript, 1 , Baseline left parenthesis, "t" , right parenthesis , right parenthesis , right parenthesis , right parenthesist+r·sinαt−sinαt+θ1t,ft+r·cosαt−cosαt+θ1t