代写MAT2355 Winter 2024代写数据结构语言程序

MAT2355

Winter 2024

• A one sheet two-sided cheat sheet is allowed.

• For full credit, you must solve 7 out of 9 of the problems in the exam.

Things to Know

1. Euclidean plane geometry: points, coordinates, distance (R2)

2. Transformations of the plane, isometries

(a) Translations, reflections, rotations (formulas for them)

(b) Composition of transformations and the groups of all isometries

(c) Writing isometries as compositions of at most 3 reflections

(d) Orientation preserving or reversing isometries.

3. Affine geometry (R and R2)

(a) general form. for affine transformations

(b) Quantities that they preserve (parallel lines, ratios of segments)

4. Affine geometry of C1

(a) general form. of affine transformations of the complex plane

(b) rotations, uniform. scalings with centers at any point in the plane

5. Projective geometry

(a) The real projective line RP1, i.e. the real line with a ’point at infinity’ added

    i. The group of projective transformations of the line

    ii. The cross-ratio of 4 points on the projective line

    iii. Finding a projective transformation which sends 3 points to 3 other points.

    iv. Harmonic conjugates (cross-ratio=−1).

    v. Constructing the 4th point in a harmonic ratio

    vi. Homogeneous coordinates

(b) The real projective plane RP2 , i.e. the real plane with a ’line at infinity’ added

    i. Homogeneous coordinates and the algebraic definition of the real projective plane

    ii. Intersections of parallel lines at infinity

    iii. Affine curves, projectivisation of curves, changing the plane in which we consider the curve

    iv. Intersections of the curves with the line at infinity

    v. (Optional) tangents and finding the asymptotes of curves, (not just the slope)

6. Projective Geometry of the complex line C1

(a) ˆC as the complex line with a point at infinity added.

(b) Generalized circles.

(c) Cross-ratios and verifying that 4 points are on a generalized circle.

(d) Geometric inversions.

(e) Finding generalized conjugates with respect to circles and/or lines.

7. Higher dimensional Euclidean Geometry

(a) Translations, orthogonal matrices

(b) Polar decomposition (for invertible matrices in dimension 2).

8. Quaternions

(a) Performing operations with quaternions.

(b) Expressing rotations in R3 using unit quaternions (one column of the matrix suffices).

9. Hyperbolic geometry

(a) The disk model D2 in the complex plane

(b) The group of isometries of D2, i.e. the Mobius transformations which preserve the unit circle and the interior of the disk

(c) Computing the hyperbolic distance between points in the disk.