# Euclid's Muse

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# Search Results for “curve”

##### Morph curve depends on the parametrization of the circle
The form of the morph curve does depend on how the original curves are parametrized. Changing the parametrization of the original curves does not change their appearance when displayed, but it does change the appearance of the morph curve.

Tags: Clocks, Morph-curve

##### Quartic Bezier Curve
An interactive model of the Quartic Bezier Curve.

Tags: Bezier, Curve, Quartic, 4th-Order, Graphics, Drawing, Vectors

##### Caustic curve as focus of parabola
The red curve is f(x) (for -2≤x≤2).  The blue curve is the terms up to the square term of the Taylor series.  A is the vertex of this parabola, B is its focus. The orange curve is the caustic - envelope of the reflection of vertical rays reflected in a mirror aligned with f(x). Observe that the focus of the parabola traces the caustic.

Tags: caustic, Taylor-series, parabola

##### Curve Area
E is the midpoint of AD and F is the midpoint of BC.  A closed curve is made by joining the cubic spline whose control polygon is EABF to the spline whose control polygon is FCDE. Can you identify a relationship between the area of the curve and the area of the polygon ABCD?

##### Morph a circle into a square
Geometrically, the morph curve is constructed by joining two corresponding points on the original curves by a line, and then positioning a third point at a specific proportion along this line.  Drag the middle red dot to see the morph curve change.

Tags: clocks, Morph-curve

##### Maximum Sum of Squares
Drag points A B and C on the curve, and try and make the sum of squares as large as possible. When you have made the number as big as you can, click Show to see the normals to the curve at A, B and C as well as the midpoints of the triangle. Can you improve your value by dragging the points so that the normals pass through the midpoints?

Tags: triangle, maximum, curve

##### Maximum Sum of Squares
Can you find positions of A, B and C on the curve which maximize (at least locally) the sum of squares of the lengths of the triangle ABC. Press Show to see the midponts of the sides of the triangles and the normals to the curve.

Tags: normals, sum-of-squares, triangle, curve

##### Minimum Perimeter Triangle
Constant force actuators pull the masses and remain tangent to the curve.  Potential energy for these actuators corresponds to their length.  Hence a minimal energy configuration should minimize the perimeter of the triangle. Press Show to see the cevians to the points of contact with the curve. Pres Show a second time to see the normals at the points of contact. What do you notice?

Tags: Cevian, triangle, curve, minimum, perimeter

##### Tangents to Polar Functions
The purple curve has polar equation r=f(θ). A lies on this curve, and B is a point at the intersection of the tangent at A with the line perpendicular to OA. The red curve is the locus of B The grey curve is the curve r=g(θ) rotated by the quantity on the slider. What function g() will rotate to lie on top of the red curve?

Tags: polar-function, tangent, spiral, Archimedes

##### Investigating Integrals through Polygons
Explore how a geometric approximation beneath the curve compares to the definite integral at variable bounds.

Tags: integral, area, area-under-the-curve, approximating-area, Riemann-sum, trapezoidal