A rotation in the plane can be formed by composing a pair of reflections. First reflect a point P to its image P′ on the other side of line L 1 . Then reflect P′ to its image P′′ on the other side of line L 2 .

Affine transformations of the plane in two dimensions include pure translations, scaling in a given direction, rotation, and shear. An affine transformation is usually and conveniently represented in matrix notation: using homogeneous coordinates. The advantage of using homogeneous coordinates is that

Given the point matrix (four points) on the right; and a line, NM, with point N at (6, -2, 0) and point M at (12, 8, 0). 3 10 1 3 . 0 0 0 0. 2. Translate N to the origin. 3. Rotate about the X axis. 4. Rotate about the Y axis. 5. Rotate 60 degree (positive) 6. Reverse [R]y. 7. Reverse [R]x. 0 …

Here we define the rotation in terms of what does change, that is the two dimensional plane within which points move as they rotate. We will factor the rotation into the following steps: Move point along perpendicular line to the plane which goes through the origin.

Jul 26, 2024 · Let $S$ be a set of $n$ points in the plane in general position (no 3 on a line), $n$ even. A halving line is a line through $2$ points of $S$ that partitions $S$ into 2 equal parts ($(n-2)/2$ points on each side). A half-set of $S$ is a subset of the form $T=S∩H$ for some halfspace $H,$ such that $|T|=n/2.$ See this image.

Start by rotating a vector x counterclockwise through an angle of 15 degrees. Recall from class that the matrix for such a rotation is R = [ cos(15 degrees) - sin(15 degrees) ]

Reflection is a special case of rotation of 180° about a line in xy plane passing through the origin. Eg about y=0 (x-axis) If two pure reflections about a line passing through the origin are applied successively the result is a pure rotation. Det[T]=? In case of a) rotation, b) reflection. What is the geometrical interpretation of inverse of. [T]?

Example: A 2D point (x,y) is the line (x,y,w), where w is any real #, in 3D homogenous coordinates. • Be sure to multiple transformations in proper order! How to interpolate keyframes? Solution: Quaternions! What about interpolating multiple keyframes? 2. Rotate about y. 3. Rotate about x. 4. Rotate about z. How do we transform among them?

Point C is called the instantaneous center of rotation. Multiplying through by ω, we have −ω × v O = ω × (ω × r ), and, re-arranging terms, we obtain, C 1 r = (ω × v O ) , C ω2 which shows that r C and v O are perpendicular, as we would expect if there is only rotation about C.

We show in this note how one can quickly rotate any figure F(x,y)=0 in 2D by any desired angle θ by simple matrix manipulations. Out starting point is to consider two orthogonal Cartesian coordinate systems [x,y] and [x',y'] having the same origin but their axis separated by angle θ from each other. Here is a schematic of the two systems,