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Linear Transformations
by Ljiljana Skrba
hey there so you wanna know how transformations work?eh?
well here i will try to explain(nothing too advanced or complicated) how to use this applet first and then more general explanations about transformations will follow!
for this applet you can use the mouse to draw lines on the graph and transform then by clicking the buttons(thro x, thro y..etc)..but REMEMBER only the last line drawn will be transformed, and then after, you can draw another, transform, and then another etc...
as for the boxes you can enter the values of x's and y's there and then press load to draw the line and then 'tranform' to transform it..but..pleeeaaaase use small numbers, 'cos the scale is from -10 -> 10!!!!!
eg.values of x: a b
values of y: c d
so your two point are : (a,c) and (b,d)
transformation for x:
coeff of x: e
coeff of y: f
so x' -> ex + fy
transformation for y:
coeff of x: g
coeff of y: h
so y' -> gx + hy
LINEAR TRANSFORMATINS:
·A transformation of a plane maps any point in the plane onto one and only one image point.(so if you put in two of the same point and have the same transformation,all that will be transformed is another point)
·If (x,y) is a point in the plane,its image point is denoted by (x',y'). This is written as f(x,y) = (x',y') .
·A linear transformation f can be expressed as a pair of equations:
x' = ax + by
y' = cx + dy, ad - bc != 0
where (x',y') is the image of (x,y) under f .
· Properties of linear Transformations:
(i) line is mapped onto a line, but the image need not be parallel to the given line.(you datermine the transformation using the 'coeffs')
(ii) a pair of parallel lines is mapped onto a pair of parallel lines.(the point have to transformed through the same transformation, i.e the 'coeffs are the same')
(iii) a parallelogram is mapped onto a parallelogram
(iv) origin is always mapped onto the origin!!!!!!!!!!!(well you are mutiplying by zero so..)
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2002
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