Search Results for “rotations”
Combination of Reflections through Oblique Lines
Set the rotation angle and drag the center of rotation O so that the orange triangle sits on top of the image under the second reflection. Can you describe another transformation equivalent to the two reflections?
Reflection Composition 2
Composition of reflections in two non-parallel lines. can you match a rotation to the combined reflection?
Reflection Clock
You can reflect the triangle in the minute hand, then reflect the reflection in the second hand. How long does it take to do a complete rotation? You can change the triangle by dragging the red points.
Fermat Toricelli Geometric Proof
We use rotations to construct a path equal in length to the sum of the distances AE+BE+CE, whose end points are fixed. Minimum for AE+BE+CE occurs when this path is a straight line. So long as no angles of the original triangle are greater than 120 degrees, this works. What goes wrong when an angle does exceed 120 degrees?
Caustic Formation
Light from C reflects in the tangent to the circle at point B. Observe the trace as B rotates around the circle. While the individual rays move, the phantom curve - the caustic - does not. The caustic is the mathematical envelope of the reflected rays, which is to say that each reflected ray is tangential to the envelope curve. At any moment, each ray is moving with a combination of rotation and translation. Each point on the ray is undergoing a translation. Points on the ray which lie on the caustic are moving along teh length of the ray, and hence their motion is not visible. Which is why the caustic looks stationary while everything else moves.
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).
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.
Geometric Top View Model of the Twisted Savonius Wind Turbine (Interactive)
This app models the top view of the Twisted Savonius Vertical Axis Wind Turbine (VAWT). The various inputs and draggable points allow you to see how the model can trace the blades' surfaces. You can also control the twist angle, radius, and rotation - which makes the whole thing spin! Learn more about the Twisted Savonius Modeling Project here.
