Question

Consider the pyramid with vertices P ₁=(−1,0,0),P ₂= (0,0,2),P₃ =(1,0,0),P 4=(0,2,0). Determine the images of these four vertices under a spin of 45 ∘

about the y-axis, followed by a tip of 30∘
about the x-axis and then an orthogonal projection into the xy-plane.

Answer 1

To find the images of the vertices, apply the spin, tip, and projection transformations step by step. Use the rotation matrices for each transformation to obtain the new coordinates. The images will have the same x and y coordinates, but the z coordinate will be zero.

Step-by-step explanation:

To find the images of the four vertices under the given transformations, we can apply each transformation step by step.

Let's start with the spin of 45 degrees about the y-axis.

This rotation will change the x and z coordinates of the vertices, while y remains the same. Using the rotation matrix for a spin about the y-axis:

| x' | | cos(45) 0 sin(45) | | x |
| y' | = | 0 1 0 | | y |
| z' | | -sin(45) 0 cos(45) | | z |

Applying this matrix multiplication to each vertex, we get the new coordinates after the spin.

Next, we apply a tip of 30 degrees about the x-axis. Similar to the previous rotation, this will change the y and z coordinates, while x remains the same. Using the rotation matrix for a tip about the x-axis:

| x' | | 1 0 0 | | x |
| y' | = | 0 cos(30) -sin(30) | | y |
| z' | | 0 sin(30) cos(30) | | z |

Again, applying this matrix multiplication to each vertex gives the new coordinates after the tip.

Finally, we need to perform an orthogonal projection into the xy-plane. This projection will remove the z coordinate and only keep the x and y coordinates.

So, the images of the four vertices under this projection will have the same x and y coordinates as before, but their z coordinate will be zero.