# Euclid's Muse

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# Search Results for “Ellipses”

##### Rotating Ellipses
Two ellipses rotate within a third such that the two smaller ellipses are always tangent to each other and the larger ellipse. Adjusting the variable X demonstrates this rotation. All three ellipses are constrained only by their implicit equations, and the moving ellipses' equations include functions of X that allow them to rotate. Can you figure out what the equations are (the other constants in the equations are a, b, h, and k)?

Tags: Ellipses, conics, puzzler

##### Rotating Ellipses with Arcs
Now that you know how two ellipses can rotate such that they are tangent to each other and a third larger ellipse (see this app); the next challenge is to figure out how an arc placed on each ellipse can be constrained (with proportional points along the ellipses) such that each arc always covers half of the ellipse and one endpoint is on the tangent point of the two smaller ellipses.

Tags: Ellipses, conics, puzzler, arcs

##### Twisted Savonius Wind Turbine Full Geometric Model (without traces/surfaces)
The Twisted Savonius Wind Turbine has promising applications for rooftop usage, but its high cost has kept it unfeasible for widespread adoption. The Twisted Savonius Geometric Modeling project explored the geometric properties of the turbine's shape, and proposed a more efficient method of construction and geometric design as a result. This is the complete side view "3d" model of the turbine. It models an extremely 3-dimensional shape by using ellipses to represent tilted circles. Changing X changes the rotation of the turbine (in operation). Theta represents the twist angle between the top and the bottom of the turbine. T controls the parametric location of the vertical surface - tracing it "fills in" the blade's surface. Learn more about this side view model. Visit the Geometry of the Twisted Savonius Wind Turbine website.

Tags: Twisted-Savonius, Wind-Turbine, Pseudo-3d, Model, Geometric, Real-World, Ellipses, Arcs, Loci

By Cannon
##### Death Star vs. Endor
An exciting mix of calculus and the hit series Star Wars, explore a problem the mighty empire could've faced with their final attempt to squash the rebellion.

Tags: Starwars, Derivatives, Normal-lines, Draggable, Fiction, Circles, Ellipses, Tangents

##### Squeezing Twisted Savonius Wind Turbine Model
This model demonstrates that the surface of the Twisted Savonius wind turbine's blades are geometrically squeezed as the twist angle is increased and the parametric position is moved up and down the turbine. Learn more about the squeeze. Learn more about the Geometry of the Twisted Savonius Wind Turbine project. Note: the calculated radius in this particular example cannot be accurate because the model is a 2d geometric approximation of the real 3d shape. Accurate calculations are made from the top view model, which is visually more difficult to comprehend. The calculation here still varies accurately as the twist angle is changed and the position is moved up and down the turbine, but it also varies as the rotation is changed (which shouldn't happen).

Tags: Twisted-Savonius, Wind-Turbine, Pseudo-3d, Model, Squeeze, Geometric, Real-World, Ellipses, Arcs, Loci, Parametric/Proportional

##### October Problem
Given two concentric similar (and similarly situated) ellipses, we create the two tangents from a point on the outer ellipse to the inner ellipse. Notice the area of the triangle formed by E and the tangency points F and G is constant as you drag E. Why? You can change the semi major axis lengths a and b, or the scale factor k.

Tags: ellipse, transform

##### Ellipse Generalization
A theorem of Archimedes on a circle can be generalized to ellipses as shown here. CE is a diameter of the ellipse. FG is parallel to the tangent at C.  H is the intersection of the tangents at F and C. Observe that EH bisects FG.

Tags: ellipse, archimedes, diameter

By Matt
##### Hex Flower
Simple flowering illustration made from a hexagon and ellipses.

Tags: Hexagon, Illustration

##### Pedal Triangles
The loci of the midpoints of the sides of pedal triangles form ellipses as the pedal point moves around a circle.

Tags: