Reflection about an arbitrary line 1.Translate the coordinates so that P 1 is at the origin 2.Rotate so that L aligns with the x-axis 3.Reflect about the x-axis 4.Rotate back 5.Translate back p 1 p 2 L = p 1 + t (p 2-p 1) = t p 2 + (1-t) p 1.
AML710 CADLECTURE 52D TRANSFORMATIONS (Contd.)Sequence of operations, Matrix multiplication, concatenation, combination of operationsTypes of Transformation Affine Map: A map φ that maps E3 into itself is called an affine Map if it leaves barycentric conditions invariant. If x = ∑ β j a j, x, a j ∈ E 3 Then, φx = ∑ β jφa j φx, φa j ∈ E 3 Most of the transformations that are used to position or scale an object in CAD are affine maps. The name was given by L. Euler and studied systematically by A. MobiusEuclidian Maps: Rigid body motions like rotation and translation where lengths and angles are unchanged are called Euclidian maps.
This is a special case of affine maps1Transformation Groups and Symmetries Affine transformations are classified (Felix Klein) as follows Similarity Groups: Rigid motion and scaling Eg: Congruent, similar Symmetry Groups:Rotation, Reflection, Translation 1. No translation and no reflection – Circle group Eg: a gear wheel 2. No translation and at least one reflection – Dihedral group 3. Translation along one direction 4. Translation along more than one directionHomogeneous coordinates of vertices ¾ A point in homogeneous coordinates (x, y, h), h ≠ 0, corresponds to the 2-D vertex (x/h, y/h) in Cartesian coordinates. ¾ Conceive that the Cartesian coordinates axes lies on the plane of h = 1.
The intersection of the plane and the line connecting the origin and (x, y, h) gives the corresponding Cartesian w coordinates. (x, y, h)y y(x/h, y/h, 1)xh=1 xh=02¾ For example, both the points (6, 9, 3) and (4, 6, 2) in the homogeneous coordinates corresponds to (2, 3) in the Cartesian coordinates. Conversely, the point (2, 1) of the Cartesian corresponds to (2, 1, 1), (4, 2, 2) or (6, 3, 3) of the homogeneous system h6, 3, 3) (4, 2, 2)y (2, 1, 1)yx xh=1 h=0Rotation Consider the following figure where a position vector p(x,y) which makes an angle φ to x-axis after transformation to p’(x’,y’) makes an angle φ+θ degrees.
Reflection of a point about a line & a plane in 2-D & 3-D co-ordinate systems by HCR.1.3-DMr Harish Chandra RajpootM.M.M. University of Technology, Gorakhpur-273010 (UP), India 12/5/2015Introduction: Here, we are interested to find out general expressions to calculate the co-ordinates of a pointwhich is the reflection of a give point about a line in 2-D co-ordinate system and about a line & a plane in 3-Dco-ordinate system as well as the foot of perpendicular to a line & a plane by using simple geometry.Reflection of a point about a line in 2-D co-ordinate system: Let there be any arbitrary point say( ) & a straight line AB:. Now, assume that the point ( ) is the reflection of thegiven point P about the given straight line AB (See the figure 1 below) then we have the following twoconditions to be satisfied1. The mid-point M of the line joining the points ( ) & ( ) must lie on the line AB2. The line joining the points ( ) & ( ) must be normal to the line ABNow, we would apply both the above conditions to find out theco-ordinates of the point ( ). Co-ordinates of the mid-point M of the line PP’ are calculated as( )The mid-point M (i.e.