Creating a Cycloid in Desmos. Equation for the cycloid that we will draw: X = sin(t), y = 1 – cos(t) 0 < t < 4 Desmos allows the user to create variables and generic functions. We will take advantage of this and create an animation that draws out the cycloid: Follow the steps below: Step 6 is YOUR task and must be completed to Step 1: Define the and variable functions. Type the following: After typing these first two functions, click on the color chooser icon to make them disappear: (We won’t need these graphs). f(t) = t * sin(t) g(t) = 1 * cos(t) Step 2: Graph the cycloid, also set the limits for t: (type “pi” for Ï€) (f(t), g(t)) 0 < t < Ï€ Step 3: Create a moving point on the cycloid: Type the following: (f(a), g(a)) Note: When you type “a” it will ask if you want to create a slider. Create the slider and set limits from 0 to 4. Move the slider around to see the animation. Step 4: Now you will create the moving center of the circle that will be “rolling” along the x-axis. (a, 1) Step 5: The “rolling” circle is really an illusion (in this animation). We will draw a circle moving along the x-axis by using the moving center point: (x – 0)^2 + (y – 1)^2 = 1 STEP 6: YOUR TASK is to create a line segment connecting the center of the circle with the point that draws out the cycloid: Recall that the line connecting points (x1, y1) and (x2, y2) is given by the parametric equations: x = x1 + (x2 – x1) y = y1 + (y2 – y1)
The Correct Answer and Explanation is:
To complete Step 6, you need to create the parametric equations for the line segment that connects the center of the rolling circle with the point on the cycloid.
Understanding the situation:
- The center of the rolling circle is at the point (a,1)(a, 1)(a,1), as you have created in Step 4.
- The point on the cycloid is at (f(a),g(a))=(a⋅sin(a),1−cos(a))(f(a), g(a)) = (a \cdot \sin(a), 1 – \cos(a))(f(a),g(a))=(a⋅sin(a),1−cos(a)), as described in Step 3.
You need to parametrize the line connecting the center (a,1)(a, 1)(a,1) and the point (f(a),g(a))(f(a), g(a))(f(a),g(a)).
Parametric Equations for the Line Segment:
Let the parametric variable ttt (which can range from 0 to 1) represent the fraction of the distance along the line between the two points. When t=0t = 0t=0, you are at the center of the circle (a,1)(a, 1)(a,1), and when t=1t = 1t=1, you are at the point on the cycloid (f(a),g(a))(f(a), g(a))(f(a),g(a)).
- X-equation: The x-coordinate for any point on the line segment will be given by: x(t)=a+t⋅(a⋅sin(a)−a)x(t) = a + t \cdot (a \cdot \sin(a) – a)x(t)=a+t⋅(a⋅sin(a)−a) Here, aaa is the fixed x-coordinate of the center, and a⋅sin(a)−aa \cdot \sin(a) – aa⋅sin(a)−a is the horizontal distance between the center and the cycloid point. The variable ttt scales this distance.
- Y-equation: The y-coordinate for any point on the line segment will be given by: y(t)=1+t⋅((1−cos(a))−1)y(t) = 1 + t \cdot ((1 – \cos(a)) – 1)y(t)=1+t⋅((1−cos(a))−1) In this case, 1 is the fixed y-coordinate of the center, and (1−cos(a))−1(1 – \cos(a)) – 1(1−cos(a))−1 is the vertical distance between the center and the cycloid point. Again, ttt scales this vertical distance.
Final Parametric Equations for the Line:
So the parametric equations for the line connecting the center to the point on the cycloid are:x(t)=a+t⋅(a⋅sin(a)−a)x(t) = a + t \cdot (a \cdot \sin(a) – a)x(t)=a+t⋅(a⋅sin(a)−a)y(t)=1+t⋅(1−cos(a)−1)y(t) = 1 + t \cdot (1 – \cos(a) – 1)y(t)=1+t⋅(1−cos(a)−1)
Conclusion:
These equations describe a line segment that starts at the center of the circle and ends at the point on the cycloid. By varying ttt from 0 to 1, you can create the effect of a line drawing from the center to the point on the cycloid as the cycloid is traced out. When combined with the rest of the cycloid animation, this will enhance the visual effect of the “rolling” circle.
