What are the different coordinate transformation conjectures?

Answer 1

Traditionally, we consider these four transformations:
Rotation, Reflection, Translation, Dilation.
However, one can invent some other types as well as a combination of them.

Rotation assumes the known center of rotation #O# and angle of rotation #phi#. The center #O# is transformed into itself. Any other point #A# on a plane can be connected with a center by a segment #OA# and the transformation rotates that segment by a given angle of rotation around point #O# (positive angle corresponds to counterclockwise rotation, negative - clockwise). The new position of the endpoint of this segment #A'# is a result of a transformation of the original point #A#.
Reflection assumes the known axis of reflection #OO'#. Any point of this axis is transformed into itself. Any other point #A# is transformed by dropping a perpendicular #AP# from it onto axis #OO'# (so, #P in OO'# is a base of this perpendicular) and extending this perpendicular beyond point #P# to point #A'# by the length equal to the length of #AP# (so, #AP=PA'#). Point #A'# is a reflection of point #A# relative to axis #OO'#.
Translation is a shift in some direction. So, we have to have a direction and a distance. These can be defined as a vector or a pair of numbers - shift #d_x# along X-axis and shift #d_y# along Y-axis. Coordinates #(x,y)# of every point are shifted by these two numbers to #(x+d_x, y+d_y)#.
Dilation is a scaling. We need a center of scaling #O# and a factor of scaling #f != 0#. Center #O# does not move anywhere by this transformation. Every other point #A# is shifted along the line #OA# connecting this point with a center #O# to another point #A' in OA# such that #|OA'|=|f|*|OA|#. Depending on the sign of factor #f#, point #A'# is positioned on the same side from center #O# on line #OA# as original point #A# (for #f>0#) or on the opposite side (for #f<0#).
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

The different coordinate transformation conjectures include:

  1. The Polar Coordinate Transformation Conjecture
  2. The Cylindrical Coordinate Transformation Conjecture
  3. The Spherical Coordinate Transformation Conjecture
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7