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In the geometry of curves, an orthoptic is the set of points for which two tangents of a given curve meet at a right angle. Examples: 1. * The orthoptic of a parabola is its directrix (proof: see ), 2. * The orthoptic of an ellipse x2/a2 + y2/b2 = 1 is the director circle x2 + y2 = a2 + b2 (see ), 3. * The orthoptic of a hyperbola x2/a2 − y2/b2 = 1, a > b, is the circle x2 + y2 = a2 − b2 (in case of a ≤ b there are no orthogonal tangents, see ), 4. * The orthoptic of an astroid x​2⁄3 + y​2⁄3 = 1 is a quadrifolium with the polar equation(see ). Generalizations:

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• Orthoptic (geometry)
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• In the geometry of curves, an orthoptic is the set of points for which two tangents of a given curve meet at a right angle. Examples: 1. * The orthoptic of a parabola is its directrix (proof: see ), 2. * The orthoptic of an ellipse x2/a2 + y2/b2 = 1 is the director circle x2 + y2 = a2 + b2 (see ), 3. * The orthoptic of a hyperbola x2/a2 − y2/b2 = 1, a > b, is the circle x2 + y2 = a2 − b2 (in case of a ≤ b there are no orthogonal tangents, see ), 4. * The orthoptic of an astroid x​2⁄3 + y​2⁄3 = 1 is a quadrifolium with the polar equation(see ). Generalizations:
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• In the geometry of curves, an orthoptic is the set of points for which two tangents of a given curve meet at a right angle. Examples: 1. * The orthoptic of a parabola is its directrix (proof: see ), 2. * The orthoptic of an ellipse x2/a2 + y2/b2 = 1 is the director circle x2 + y2 = a2 + b2 (see ), 3. * The orthoptic of a hyperbola x2/a2 − y2/b2 = 1, a > b, is the circle x2 + y2 = a2 − b2 (in case of a ≤ b there are no orthogonal tangents, see ), 4. * The orthoptic of an astroid x​2⁄3 + y​2⁄3 = 1 is a quadrifolium with the polar equation(see ). Generalizations: 1. * An isoptic is the set of points for which two tangents of a given curve meet at a fixed angle (see ). 2. * An isoptic of two plane curves is the set of points for which two tangents meet at a fixed angle. 3. * Thales' theorem on a chord PQ can be considered as the orthoptic of two circles which are degenerated to the two points P and Q.
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