Day: October 12, 2023

Posted on

202310121132 Pastime Exercise 005

The blogger claims no originality of his problem below.

Below is a \textrm{3-D} graph of the customary quadratic equation on the flat xy-plane at some level of z=d:

where the discriminant is \Delta =b^2-4ac.

Setup.

Let ten operators \hat{Q}_1, \hat{Q}_2, \hat{Q}_3, \hat{Q}_4, \hat{Q}_5, \hat{Q}_6, \hat{Q}_7, \hat{Q}_8, \hat{Q}_9, and \hat{Q}_{10} be defined as such that follow:

Of the curve f(x,y,z)\big|_{z=d}=0,

\hat{Q}_1: a horizontal translation along (\pm\textrm{ve})\, x-direction;
\hat{Q}_2: a vertical translation along (\pm\textrm{ve})\, y-direction.
\hat{Q}_3: a horizontal scaling by a (\pm\textrm{ve}) factor;
\hat{Q}_4: a vertical scaling by a (\pm\textrm{ve}) factor;
\hat{Q}_5: a(n) anti-/clockwise rotation about a normal on the xy-plane;
\hat{Q}_6: a horizontal reflection over a vertical line parallel to the y-axis;
\hat{Q}_7: a vertical reflection over a horizontal line parallel to the x-axis;
\hat{Q}_8: an oblique reflection over a slanted line on the xy-plane;
\hat{Q}_9: a tilting about the (\pm\textrm{ve})\,y-axis;
\hat{Q}_{10}: a tilting about the (\pm\textrm{ve})\,z-axis.

Problem.

(a) Describe the ten operators explicitly in the form of some linear functions.

(b) Does it matter to have one operator performed with some priority over another in order to give a curve and the only curve after multiple transformations? If you think so, list them in order; if not, give a counterexample that any two sequences of execution are able to reach the same result.

This problem is not to be attempted.