Shear Transformation – 2D Transformation – Computer Aided Design

Shear Transformation – 2D Transformation – Computer Aided Design


Hello friends today we will start one more special type of transformation that is shear transformation so shear transformation is slanting of the image means the image of the object will slide so first I will write the definition for shear transformation [Music] so a Shear transformation that is a transformation that distorts the shape and size of the object which it will distort the shape and size of the object such that the object which we’ll call to slide over each other means if we will apply the shear transformation the object will get tilted which is also called as slanting of the image so it can also be said that transformation which means slanting of the image is called a shear transformation so I can also say it as slanting of the image so shear transformation are of two types that is first X here second is why shell which means that she can takes place along X direction as well as log Phi direction so now we will study X here [Music] now we will derive the matrices that is resultant matrices for shear transformation which is along X direction so now I will draw in figure [Music] so this is the figure that is before X here when I will apply texture X share means what we will slide or we will steal the image along X direction so now I will apply a shear transformation which means that I am applying shear along X direction so now the figure will be transformed as so now the after X shear I will name it as a – B – C – Andy – which means that I am giving us – for after shear so I will know it as a dash B dash C dash and D dash so the thinking of rectangle implies the figure before X shear and red color rectangle which I am represented in – shows the figure after H shear transformation you can see the y coordinate value is unchanged you can see the example that is there – and a which means that it is on the same line that is it is an origin and in the case of B dash and B you can see that this D one first D which is of original figure and after a player definition it has named as B dash which says that it is on same point the shear transformation has now occurred along y direction it is occurred along x direction so so the Y value remains unchanged but in the case of X here you can see originally it was d and after applying shared or information it is transformed into D – this is at point so you can see that Oh it wasn’t y-axis after oblem sheer the value has been changed so now D has occupied this position similarly you can see for C this is C after a plane shear transformation the sea has been transformed into see – you can see that the value of C in Rosia and after uplink shared observation it has been transformed into see – you can see that since it is moving on right side the value has changed so x value causes [Music] vertical line to tilt you can see that this vertical line has tilted up to apply X here so we’ll consider one point I will name it as p1 whose coordinate is X comma Y which means that y coordinate value of p2 is unchanged and AZ coordinate value of p2 is change so now we will derive the transformation matrix for X here [Music] I limit as shx which means that shear along X direction so the matron says for shear along a direction is 1 0 0 a 1 0 0 0 1 so this is the matrices for shear along X direction where air implies shearing parameter which will show how much to tint or how much to slant so now we will derive the final position we know that final position is equal to initial x SS X because the transformation we are is sheer transformation along X direction so where we took point X comma Y now we will convert that point coordinate into matrices form so we know that the conversion of coordinate into matrices form as X y 1 so now the SHS that is shear transformation matrix is 1 0 0 a 1 0 0 0 1 so after multiplying the initial position with shx the matrices which we get [Music] so after multiplying both the matrices I will get as X plus y express Y into a comma Y comma 1 so this is this final position so now we will derive the shear matrices for y that is shear along y direction [Music] that is why shear I will draw the figure for why shear the initial figure I am drawing with Lupin so I’ll aim it as a b c d and the final figure I am drawing with red pen from the name itself why share means will apply shear along y direction which means that in this direction so now the figure will become in this way that is B will be transformed into B dash and C will be transformed into three – and if you let a since it is unchanged so I will date as a dash and D dash so from the figure it is clear that Y shear is taking place which means that the value of B which y-coordinate value of be transformed into B dash and y-coordinate of value C as will transform into C – so I can write as x coordinate value remains unchanged because the shear is taking in a Y direction so there is no change in the value along x coordinate but the y coordinate value causes [Music] horizontal line to tilt one you can see initially this a B and C D was horizontally after applying Y shear it has been transformed into a dash B dash you can see horizontal one which is tilted which has been transformed into tilting one so now we will derive the shear matrices for Y so the matrix is for shear along y direction is given as SH Y 1 B 0 0 1 0 0 0 1 this base the shearing parameter which will decide how much to tilt so now we find the final position we know that the formula for deriving Phi V position is initial position into digital transformation or transformation which we had been here [Music] final position is Alicia into SH Y so now we’ll consider point P whose coordinate is X comma Y so I will convert that coordinate into matrices form so it will become x y1 and we know the matrices for shear along y direction is 1 be 0 0 1 0 0 0 1 so now after multiplying this matrices with this I will get the file position to the final position which will become as X into 1 that is X Y into 0 is 0 over root of 0 is 0 which means that here it is X now X into B which is X into B plus y into 1 and last row is 0 so X plus y into B and last it is always as written as 1 so now the final position we got X X plus YB so this is the final position matrices thank you

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