Rotation assumes the known center of rotation OO and angle of rotation phiϕ. The center OO is transformed into itself. Any other point AA on a plane can be connected with a center by a segment OAOA and the transformation rotates that segment by a given angle of rotation around point OO (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).
