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    Laminae of Constant Width

    Named laminae

    annulus | disk | Reuleaux triangle

    Defining inequalities

    b^2<=x^2 + y^2<=a^2

    x^2 + y^2<=a^2

    (x - a/2)^2 + (a/(2 sqrt(3)) + y)^2<=a^2 and (y - a/sqrt(3))^2 + x^2<=a^2 and (a/2 + x)^2 + (a/(2 sqrt(3)) + y)^2<=a^2

    Definitions

    Lamina properties

    annulus | 00 Reuleaux triangle | a>0

    annulus | A = π (a^2 - b^2) disk | A = π a^2 Reuleaux triangle | A = 1/2 (π - sqrt(3)) a^2

    annulus | x^_ = (0, 0) disk | x^_ = (0, 0) Reuleaux triangle | x^_ = (0, 0)

    Mechanical properties

    annulus | J_x invisible comma x = 1/4 π (a^4 - b^4) disk | J_x invisible comma x = (π a^4)/4 Reuleaux triangle | J_x invisible comma x = 1/48 (10 π - 17 sqrt(3)) a^4

    annulus | J_y invisible comma y = 1/4 π (a^4 - b^4) disk | J_y invisible comma y = (π a^4)/4 Reuleaux triangle | J_y invisible comma y = 1/48 (10 π - 17 sqrt(3)) a^4

    annulus | J_zz = 1/2 π (a^4 - b^4) disk | J_zz = (π a^4)/2 Reuleaux triangle | J_zz = 1/24 (10 π - 17 sqrt(3)) a^4

    annulus | J_x invisible comma y = 0 disk | J_x invisible comma y = 0 Reuleaux triangle | J_x invisible comma y = 0

    annulus | r_x = 1/2 sqrt(a^2 + b^2) r_y = 1/2 sqrt(a^2 + b^2) disk | r_x = a/2 r_y = a/2 Reuleaux triangle | r_x = 1/2 sqrt(5/3 + 7/(6 - 2 sqrt(3) π)) a r_y = 1/2 sqrt(5/3 + 7/(6 - 2 sqrt(3) π)) a

    annulus | K = 1/2 π (a^4 - b^4) disk | K = (π a^4)/2

    Distance properties

    annulus | p = 2 π (a + b) disk | p = 2 π a Reuleaux triangle | p = π a

    disk | r = a Reuleaux triangle | r = (1 - 1/sqrt(3)) a

    disk | R = a Reuleaux triangle | R = a/sqrt(3)

    annulus | 2 a disk | 2 a Reuleaux triangle | a

    annulus | χ = (2 (a^2 cos^(-1)(b/a) - b sqrt(a^2 - b^2)))/(π (a^2 - b^2)) disk | χ = 1 Reuleaux triangle | χ = 1

    disk | s^_ = (128 a)/(45 π)

    disk | A^_ = (35 a^2)/(48 π^2)

    Common classes

    laminae of constant width

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