Rotation of an object around a point by angle is first used on the tire spokes to simulate movement. Line segments are rotated around the center point of the circle by angle β. The angle can be altered by sliderβ, and shows how increasing the angle moves the spokes in a counter-clockwise direction, while decreasing the angle causes a clockwise rotation.
The sun is also rotated around a point N, to show its movement across the sky. Its rotation is controlled by angle θ. Again, to show clockwise rotation, the angle is set in animation to move in a negative, or rather, decreasing direction. The rotation can also be controlled outside of the animation by slider θ. The rotation of the sun also controls the angle of a vector used to alter the shadow of the bike. This alteration is shown by checking “Translation by Vector”. By moving slider θ, the user can see the effect of the sun’s position on the bike’s shadow.
Reflection of an object about a line or point is used to create the shadow of the bike. I used two checkboxes to demonstrate different properties of reflection. By checking “Reflection-Perpendicular Bisector Property”, the user can see that under the definition of reflection, the line of reflection becomes the perpendicular bisector of the line segment between B and its reflection B’. By checking “Triangle Reflection Properties, the user can see by the reflection of Triangles O’AM’ to O”A’M’’ that the properties of reflection hold: by reflection, angle measure is preserved, segment length is preserved and orientation of triangle O’AM’ is changed from clockwise, to counterclockwise for triangle O”A’M”.
Dilation is used to make the flower appear to grow. The polygon is dilated by a factor controlled by slider o, from the point at the bottom of the stem. By checking “Dilation”, the user can verify the properties of dilation. By comparing segment length e3 to its dilation o8, the user can see that the ratio o8/e3 is equal to the factor on slider o, and that angle measure is preserved by comparing angle κ and its dilation, angle λ .
Translation by vector was used to alter the position of the bike reflection, allowing the position of the sun, and the length of the vector, to control the position of the shadow. By checking “Translation by Vector” checkbox, Vector u translates the position of the tires’ reflection or shadow by the length and angle of the vector. As stated above, the angle of the vector is controlled by slider θ for the sun, and the user can change the length of the vector by moving point N1.
