webwork-open-problem-library/Contrib/BentleyUniversity/MathematicsOfComputerGraphics/setMoCG-Ch5/5-1-ConvertToHomogeneous.pg at 5b81d053600776c4338fd7c5c39927c53f8d10da · openwebwork/webwork-open-problem-library · GitHub

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## DESCRIPTION
## Converting points and vectors to homogeneous coordinates
## ENDDESCRIPTION

## DBsubject(Geometry)
## DBchapter(Vector geometry)
## DBsection(Coordinate systems)
## Date(02/02/2018)
## Institution(Bentley University)
## Author(Nathan Carter)
## TitleText1('Introduction to the Mathematics of Computer Graphics')
## AuthorText1('Nathan Carter')
## EditionText1('1')
## Section1('5')
## Problem1('1')
## KEYWORDS('homogeneous coordinates')

DOCUMENT();
loadMacros(
  "PGstandard.pl",
  "MathObjects.pl",
  "PGML.pl",
  "PGcourse.pl",

);

Context("Matrix");
$showPartialCorrectAnswers = 1;
@r = (
    non_zero_random(-10,10),
    non_zero_random(-10,10),
    non_zero_random(-10,10),
    non_zero_random(-10,10),
    non_zero_random(-10,10),
    non_zero_random(-10,10),
    non_zero_random(-10,10),
    non_zero_random(-10,10),
    non_zero_random(-10,10),
    non_zero_random(-10,10),
);
$ans1 = Matrix( [
    [ $r[0] ],
    [ $r[1] ],
    [ 1 ]
] );
$ans2 = Matrix( [
    [ $r[2] ],
    [ $r[3] ],
    [ 0 ]
] );
$ans3 = Matrix( [
    [ $r[4] ],
    [ $r[5] ],
    [ $r[6] ],
    [ 1 ]
] );
$ans4 = Matrix( [
    [ $r[7] ],
    [ $r[8] ],
    [ $r[9] ],
    [ 0 ]
] );

TEXT(beginproblem());
BEGIN_PGML
Express [`([$r[0]],[$r[1]])`] in homogeneous coordinates.

[_____]*{$ans1} [@ helpLink("matrices") @]*

Express [`\langle [$r[2]],[$r[3]] \rangle`] in homogeneous coordinates.

[_____]*{$ans2} [@ helpLink("matrices") @]*

Express [`([$r[4]],[$r[5]],[$r[6]])`] in homogeneous coordinates.

[_____]*{$ans3} [@ helpLink("matrices") @]*

Express [`\langle [$r[7]],[$r[8]],[$r[9]] \rangle`] in homogeneous coordinates.

[_____]*{$ans4} [@ helpLink("matrices") @]*
END_PGML

BEGIN_PGML_SOLUTION
A point [`(x,y)`] in homogeneous coordinates is the column vector [`\left[\begin{array}{c} x \\ y \\ 1 \end{array}\right]`].

A vector [`\langle x,y \rangle`] in homogeneous coordinates is the column vector [`\left[\begin{array}{c} x \\ y \\ 0 \end{array}\right]`].

A point [`(x,y,z)`] in homogeneous coordinates is the column vector [`\left[\begin{array}{c} x \\ y \\ z \\ 1 \end{array}\right]`].

A vector [`\langle x,y,z \rangle`] in homogeneous coordinates is the column vector [`\left[\begin{array}{c} x \\ y \\ z \\ 0 \end{array}\right]`].
END_PGML_SOLUTION

ENDDOCUMENT();