Math 621: Elementary Geometry |
1. Derive the matrix representation of a rotation by \(\theta\) radians
about the origin; \(R(O,\theta)\).
2. Derive the matrix represention of a reflection across a line \(\ell\)
that passes through the origin; \(R(\ell)\), \(\ell: y = mx\) or \(x = 0\).
3. Derive the matrix representation of a homotethy by \(k\) centered at
the origin; \(H(O,k)\), \(k \neq 0\).
4. Construct the image of a triangle \(\Delta ABC\) under each of the
types of transformations in section 4.1. Focus especially on relfections,
rotations, translations, and homotheties.
5. Construct the interior and exterior centers of similitude of two
non-concentric circles of different radii.
6. Given a circle \(\Sigma_1 = O(r)\) and a point \(P\) outside of the
circle, construct a new circle \(\Sigma_2\) that is the image of \(\Sigma_1\)
under the homothety \(H(P,k)\), where \(k = \dfrac{\bar{PO}}{r}\). Find another
point \(Q\) on the line \(PO\) for which \(\Sigma_2\) is the image of
\(\Sigma_1\) under the homothety \(H(Q,-k)\).
7. Prove 1.) the composition of two isometries is an isometry; 2.) if
\(f\) is an isometry then \(f^{-1}\) is an isometry.
8. Let \(y = f(x)\) be a smooth function, and let \(P(x_0,f(x_0))\) be a
point on its graph. Describe the transformation that carries the \(x\)-axis to
the tangent line at \(P.\)
9. A set of points \(S\) in the plane is said to be invariant under
a transformation \(L\) if and only if \(L(S) = S\). Describe the transformation(s)
of the plane under which the graphs of even and odd functions are invariant.
10. Given an angle \(\measuredangle ABC = \theta\) and a point
\(P\) in the plane, the transformation \(R(P,\theta)\) is an isometry. Is it
orientation preserving or reversing? Construct the lines \(\ell_i\) satisfying
\(R(P,\theta) = \prod R(\ell_i)\).
11. Given two congruent triangles \(\Delta ABC\) and \(\Delta A'B'C'\),
find the transformations \(R(\ell_i)\) that form the isometry mapping one of
the triangles to the other. (Use these pictures: Sample 1, Sample 2, Sample 3 )
12. Suppose a linear transformation of the plane can be represented by a
matrix that has a single eigenvalue 3 of multiplicity 2. Describe the
transformation; i.e., classify it as a composition of the transformation
types in section 4.1.
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