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Theorem: If one vertex of movably pivoted rhombus is constrained to move in the circumference of a directing circle, while the opposite vertex is fixed in the diameter(or diameter produced), the locus of the intersection of the diagonal (produced) through the other two vertices with the radius of the directing circle is a conic. |
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Pf:
Let BIDF be the rhombus with the
vertex B moving in the circumference of the directing circle whose center
is F'; F the opposite fixed within (without )the diameter ; and DI the
diagonal produced to intersect the radius (produced) in the point K. Draw KF, and FK’. F'K－FK＝F'K－BK＝BF'＝constant. ∴The locus of the K is an hyperbola. In the figure to the middle, FK＋KF' ＝BK＋KF'＝BF'＝constant. ∴The locus of the K is an ellipse. In the figure to the left, suppose that the radius of the directing circle becomes infinite. Then the circumference becomes the line XY, perpendicular to the diameter ; BF becomes the line B'F'', parallel to the diameter , and the diagonal D'I' intersects it in K'.Hence B'K' ＝FK'. ∴The locus of the K' is an parabola.
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| Use for linkage: hyperbola If B is moves in the circumference of a directing circle, and the point F is fixed outside the circle F', then the point K will describe a hyperbola as the linkage move. |
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| Use for linkage: hyperbola If B is moves in the circumference of a directing circle, and the point F is fixed outside the circle F', then the point K will describe a hyperbola as the linkage move.
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| Use for linkage: Parabola If B' is moves in the line L, and the point F" is fixed on a line perpendicular to L, then the point K will describe a hyperbola as the linkage move.
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| Reference:
A linkage for Describing the Conic Sections by Continuous Motion , |
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