Euclids Elements - Book 1 - Proposition 45

Creating a parallelogram equal to a given quadrilateral with a given angle.

Tags: Euclid, Elements, Geometry, Parallelogram, Triangle, Angle

Projectile High Point

Press the play button to fire a projectile in the direction of the arrow. Move the arrow and try again. Notice the high points of the trajectory lie on an ellipse.  If the initial velocity of the projectile is v, and gravity is g, what is the equation of the ellipse?

Tags: projectile, parabola, ellipse, physics

Lissajous Figure

These were (easy and) fun to create on oscilloscopes in the dark days before computers.

Tags: Lissajous, parametric-curve

Hyperbola Polar Line

The polar line is the locus of the intersections of tangent lines at the ends of chords of teh parabola through a fixed point. Turns out to be conceptually important - not just a curiosity.

Tags: polar, conic, hyperbola

hyperbolic clock with explanation

The hyperbolic clock is all about the parametrization.  Press the explain button to see why it moves the way it does.

Tags: hyperbola, clock

Fourier Series

Try different values for the parameters, to generate different waveforms.

Tags: Fourier, waveform, trigonometry

Cam with reciprocating roller follower

Alter the parameters of the output curve and follower geometry, and see the cam shape change.

Tags: cam, sine

Spline Approximation

Spline approximation to a quarter unit circle.  Can you find the best value for parameter a?

Tags: spline, circle

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