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Questions and Answers
What are the approximate vertices of a triangle after rotating 180 degrees counterclockwise about the point (1, 1)?
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) (correct)
- (-1, 0), (0, -2), and (1, 1)
- (2, 0), (0, 2), and (1, 1)
- (0, 0), (2, 0), and (1, -2)
When rotating a point (2, 3) by 90 degrees counterclockwise about the origin, what is the new x-coordinate?
When rotating a point (2, 3) by 90 degrees counterclockwise about the origin, what is the new x-coordinate?
- 2
- -3 (correct)
- -2
- 3
What is the effect of rotating a shape by 180 degrees about a point?
What is the effect of rotating a shape by 180 degrees about a point?
- It flips the shape across a specified axis.
- It preserves orientation, but changes shape.
- It reduces the size of the shape.
- It preserves shape and size, but changes orientation. (correct)
What is the new position of a point (2, 3) after rotating it by 45 degrees clockwise about the point (3, 3)?
What is the new position of a point (2, 3) after rotating it by 45 degrees clockwise about the point (3, 3)?
What happens to the size and shape of an object after a rotation?
What happens to the size and shape of an object after a rotation?
What is the effect of rotating a shape by 270 degrees counterclockwise about the origin?
What is the effect of rotating a shape by 270 degrees counterclockwise about the origin?
What is the result of shearing a point (2, 3) along the x-axis by a factor of 2?
What is the result of shearing a point (2, 3) along the x-axis by a factor of 2?
What is the purpose of 2D viewing in computer graphics?
What is the purpose of 2D viewing in computer graphics?
What is the result of rotating a point (2, 3, 4) by angles (45, 30, 60) degrees?
What is the result of rotating a point (2, 3, 4) by angles (45, 30, 60) degrees?
What is the result of reflecting a point (2, 3, 4) about the xy-plane?
What is the result of reflecting a point (2, 3, 4) about the xy-plane?
What is the result of translating a point (2, 3, 4) by (1, -2, 3)?
What is the result of translating a point (2, 3, 4) by (1, -2, 3)?
What is the result of scaling a point (2, 3, 4) by factors (2, 0.5, 3)?
What is the result of scaling a point (2, 3, 4) by factors (2, 0.5, 3)?
What is the effect of rotating a point by 45 degrees clockwise around the origin?
What is the effect of rotating a point by 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?
If a triangle is rotated by 90 degrees counter-clockwise around the origin, what is the resulting orientation of the triangle?
A square is rotated by 180 degrees around the origin. What is the resulting orientation of the square?
A square is rotated by 180 degrees around the origin. What is the resulting orientation of the square?
What is the effect of rotating a circle by 45 degrees clockwise around its center?
What is the effect of rotating a circle by 45 degrees clockwise around its center?
If a rectangle is rotated by 90 degrees counter-clockwise around the origin, what is the resulting orientation of the rectangle?
If a rectangle is rotated by 90 degrees counter-clockwise around the origin, what is the resulting orientation of the rectangle?
A triangle is rotated by 135 degrees clockwise around the origin. What is the resulting orientation of the triangle?
A triangle is rotated by 135 degrees clockwise around the origin. What is the resulting orientation of the triangle?
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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)
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