18 Questions
What are the approximate vertices of a triangle after rotating 180 degrees counterclockwise about the point (1, 1)?
(0.71, 1.41), (-1.41, 0.71), and (0, 2.12)
When rotating a point (2, 3) by 90 degrees counterclockwise about the origin, what is the new x-coordinate?
-3
What is the effect of rotating a shape by 180 degrees about a point?
It preserves shape and size, but changes orientation.
What is the new position of a point (2, 3) after rotating it by 45 degrees clockwise about the point (3, 3)?
(2.41, 1.41)
What happens to the size and shape of an object after a rotation?
The size remains the same, and the shape remains the same.
What is the effect of rotating a shape by 270 degrees counterclockwise about the origin?
It preserves shape and size, but changes orientation.
What is the result of shearing a point (2, 3) along the x-axis by a factor of 2?
(4, 3)
What is the purpose of 2D viewing in computer graphics?
To translate and transform graphical primitives from world coordinates into screen coordinates
What is the result of rotating a point (2, 3, 4) by angles (45, 30, 60) degrees?
A point with different coordinates in the xy-plane
What is the result of reflecting a point (2, 3, 4) about the xy-plane?
A point with the same x and y coordinates, but a negative z coordinate
What is the result of translating a point (2, 3, 4) by (1, -2, 3)?
(3, 1, 7)
What is the result of scaling a point (2, 3, 4) by factors (2, 0.5, 3)?
(4, 1.5, 12)
What is the effect of rotating a point by 45 degrees clockwise around the origin?
The point moves 45 degrees clockwise around the origin
If a triangle is rotated by 90 degrees counter-clockwise around the origin, what is the resulting orientation of the triangle?
The triangle is rotated 90 degrees counter-clockwise around the origin
A square is rotated by 180 degrees around the origin. What is the resulting orientation of the square?
The square is unchanged
What is the effect of rotating a circle by 45 degrees clockwise around its center?
The circle is unchanged
If a rectangle is rotated by 90 degrees counter-clockwise around the origin, what is the resulting orientation of the rectangle?
The rectangle is rotated 90 degrees counter-clockwise around the origin
A triangle is rotated by 135 degrees clockwise around the origin. What is the resulting orientation of the triangle?
The triangle is rotated 45 degrees counter-clockwise around the origin
Study Notes
Transformations
- Rotation: changes the orientation of an object around a fixed point or axis
- Reflection: flips an object across a specified axis, creating a mirrored image
- Shearing: distorts the shape of an object by shifting its coordinates along one axis
Rotation
- Rotating a point (2, 3) by 90 degrees counterclockwise about the origin results in vertices approximately (0.71, 1.41)
- Formula for rotating a point (x, y) by angle θ about the origin: (x cos θ - y sin θ, x sin θ + y cos θ)
Reflection
- Reflecting a point (2, 3) about the x-axis results in (-2, 3)
- Reflecting a line segment about the y-axis results in a mirrored image
- Reflection can be achieved by negating the coordinates along the axis of reflection
Shearing
- Shearing a rectangle horizontally by a factor of 2 results in new vertices
- Formula for shearing a point (x, y) along the x-axis by a factor of k: (x + ky, y)
3D Transformation
- Translate a point (2, 3, 4) by (1, -2, 3) results in (3, 1, 7)
- Scale a point (2, 3, 4) by factors (2, 0.5, 3) results in (4, 1.5, 12)
- Rotate a point (2, 3, 4) by angles (45, 30, 60) degrees
- Reflect a point (2, 3, 4) about the xy-plane
- Shear a point (2, 3, 4) along the z-axis by a factor of 2
2D Viewing
- 2D viewing is the process of rendering two-dimensional scenes or objects onto a two-dimensional display surface
- Primary purpose is to translate and transform graphical primitives (points, lines, and polygons) from their world coordinates into screen coordinates for display
Translation
- Translating a point (3, 4) by (2, -3) results in (5, 1)
- Formula for translating a point P(x, y, z) by ∆x, ∆y, and ∆z: T(x, y, z) = (x + ∆x, y + ∆y, z + ∆z)
Scaling
- Scaling a rectangle by a factor of 2 results in new vertices
- Formula for scaling a point (x, y) by a factor of k: (kx, ky)
Practice your geometry skills with these rotation problems. Rotate points, lines, triangles, and squares by different angles about specific points. Test your understanding of geometric transformations.
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