Rigid Transformations
Rigid transformations are fundamental concepts in geometry and computer science, involving the movement and positioning of objects in space while preserving their shape and size. These transformations include translations, rotations, and reflections, which maintain the distances and angles between points. They are extensively used in fields like computer graphics, robotics, and engineering for tasks such as modeling physical movements, aligning images, and designing mechanical systems. Understanding rigid transformations is crucial for accurately representing and manipulating objects in both virtual and physical environments, contributing to advancements in simulation, visualization, and automation technologies.
Questions
- Circle A has a radius of #3 # and a center of #(2 ,1 )#. Circle B has a radius of #2 # and a center of #(7 ,8 )#. If circle B is translated by #<4 ,2 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- A line segment has endpoints at #(2 , 2)# and #(5 , 4)#. If the line segment is rotated about the origin by #(pi)/2 #, translated horizontally by #1#, and reflected about the y-axis, what will the line segment's new endpoints be?
- Circle A has a radius of #5 # and a center of #(2 ,6 )#. Circle B has a radius of #2 # and a center of #(4 ,3 )#. If circle B is translated by #<-3 ,1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- A line segment has endpoints at #(6 ,5 )# and #(5 ,7)#. If the line segment is rotated about the origin by #pi /2 #, translated vertically by #2 #, and reflected about the x-axis, what will the line segment's new endpoints be?
- A line segment has endpoints at #(2 ,6 )# and #(7 , 3 )#. If the line segment is rotated about the origin by # pi /2 #, translated horizontally by # 1 #, and reflected about the x-axis, what will the line segment's new endpoints be?
- Circle A has a radius of #3 # and a center of #(3 ,2 )#. Circle B has a radius of #1 # and a center of #(4 ,7 )#. If circle B is translated by #<2 ,-3 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- A line segment has endpoints at #(7 ,4 )# and #(5 ,2)#. If the line segment is rotated about the origin by #(3pi )/2 #, translated vertically by #5 #, and reflected about the y-axis, what will the line segment's new endpoints be?
- A line segment has endpoints at #(9 ,4 )# and #(1 , 8 )#. If the line segment is rotated about the origin by # pi /2 #, translated horizontally by # 5 #, and reflected about the y-axis, what will the line segment's new endpoints be?
- A line segment has endpoints at #(2 ,3 )# and #(3 ,9 )#. If the line segment is rotated about the origin by #( pi)/2 #, translated vertically by #-8 #, and reflected about the x-axis, what will the line segment's new endpoints be?
- A line segment has endpoints at #(8 , 4)# and #(1 , 2)#. If the line segment is rotated about the origin by #(pi)/2 #, translated vertically by #4#, and reflected about the x-axis, what will the line segment's new endpoints be?
- A line segment has endpoints at #(2 , 3)# and #(1 , 2)#. If the line segment is rotated about the origin by #(pi)/2 #, translated vertically by #3#, and reflected about the x-axis, what will the line segment's new endpoints be?
- A line segment has endpoints at #(9 ,1 )# and #(5 ,3)#. If the line segment is rotated about the origin by #pi #, translated vertically by #-2 #, and reflected about the x-axis, what will the line segment's new endpoints be?
- A line segment has endpoints at #(7 ,1 )# and #(7 ,5 )#. If the line segment is rotated about the origin by # pi #, translated horizontally by # - 2 #, and reflected about the x-axis, what will the line segment's new endpoints be?
- A line segment has endpoints at #(2 ,3 )# and #(6 ,5)#. If the line segment is rotated about the origin by #(3pi )/2 #, translated horizontally by #-1 #, and reflected about the x-axis, what will the line segment's new endpoints be?
- A line segment has endpoints at #(1 ,5 )# and #(0 ,1 )#. If the line segment is rotated about the origin by # pi #, translated horizontally by # - 4 #, and reflected about the y-axis, what will the line segment's new endpoints be?
- Circle A has a radius of #3 # and a center of #(2 ,7 )#. Circle B has a radius of #2 # and a center of #(6 ,1 )#. If circle B is translated by #<2 ,7 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- A line segment has endpoints at #(7 ,5 )# and #(6 ,5)#. If the line segment is rotated about the origin by #(3pi )/2 #, translated vertically by #3 #, and reflected about the x-axis, what will the line segment's new endpoints be?
- A line segment has endpoints at #(8 ,1 )# and #(7 ,5 )#. If the line segment is rotated about the origin by # pi #, translated horizontally by # - 2 #, and reflected about the x-axis, what will the line segment's new endpoints be?
- A line segment has endpoints at #(4 ,0 )# and #(2 ,9 )#. If the line segment is rotated about the origin by #( pi)/2 #, translated vertically by #-8 #, and reflected about the x-axis, what will the line segment's new endpoints be?
- Circle A has a radius of #2 # and a center of #(1 ,7 )#. Circle B has a radius of #2 # and a center of #(8 ,1 )#. If circle B is translated by #<-4 ,3 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?