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    Tangent Line Circle

    Results

    line | T^^(t) = (b/sqrt(a^2 + b^2), -a/sqrt(a^2 + b^2)) circle | T^^(t) = (-sin(t), cos(t))

    (b/sqrt(a^2 + b^2) | -a/sqrt(a^2 + b^2) -sin(t) | cos(t))

    Example plots

    Example plots Line

    Example plots Circle

    Equations

    line | x(t) = b t y(t) = a (-t) - c/b circle | x(t) = a cos(t) y(t) = a sin(t)

    line | a x + b y + c = 0 circle | x^2 + y^2 = a^2

    line | r(θ) = -c/(a cos(θ) + b sin(θ)) circle | r(θ) = a

    Characteristic polynomial

    -(b λ)/sqrt(a^2 + b^2) - (a sin(t))/sqrt(a^2 + b^2) + (b cos(t))/sqrt(a^2 + b^2) + λ^2 - λ cos(t)

    Eigenvalues

    λ_1 = (sqrt(a^2 + b^2) cos(t) - sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) + b)/(2 sqrt(a^2 + b^2))

    λ_2 = (sqrt(a^2 + b^2) cos(t) + sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) + b)/(2 sqrt(a^2 + b^2))

    Eigenvectors

    v_1 = (-(-sqrt(a^2 + b^2) cot(t) + b csc(t) - csc(t) sqrt((4 a sqrt(a^2 + b^2) - 2 b sqrt(a^2 + b^2) cot(t) + a^2 cos(t) cot(t) + b^2 cos(t) cot(t) + b^2 csc(t)) sin(t)))/(2 sqrt(a^2 + b^2)), 1)

    v_2 = (-(-sqrt(a^2 + b^2) cot(t) + b csc(t) + csc(t) sqrt((4 a sqrt(a^2 + b^2) - 2 b sqrt(a^2 + b^2) cot(t) + a^2 cos(t) cot(t) + b^2 cos(t) cot(t) + b^2 csc(t)) sin(t)))/(2 sqrt(a^2 + b^2)), 1)

    Diagonalization

    (b/sqrt(a^2 + b^2) | -a/sqrt(a^2 + b^2) -sin(t) | cos(t)) = S.J.S^(-1) where S = ((sqrt(a^2 + b^2) cot(t) + csc(t) sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) - b csc(t))/(2 sqrt(a^2 + b^2)) | cot(t)/2 - (csc(t) (sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) + b))/(2 sqrt(a^2 + b^2)) 1 | 1) J = ((sqrt(a^2 + b^2) cos(t) - sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) + b)/(2 sqrt(a^2 + b^2)) | 0 0 | (sqrt(a^2 + b^2) cos(t) + sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) + b)/(2 sqrt(a^2 + b^2))) S^(-1) = ((sqrt(a^2 + b^2) sin(t))/sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) | (-sqrt(a^2 + b^2) cos(t) + sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) + b)/(2 sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2)) -(sqrt(a^2 + b^2) sin(t))/sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) | (sqrt(a^2 + b^2) cos(t) + sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) - b)/(2 sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2)))

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