I have an image with a bigger scale in Y than in X. In this image I want to measure several areas. Now I need to know, if the true area is dependent of the orientation of the object / area I want to measure.
Look at my example:
A is the original picture. All bars are of the same size and area.
B is picture A with a bigger height (Y) scaling
Do all three bars (horizontally vs. vertically vs. diagonally) in B have the same area?
So to summarize my question again: A picture with objects of the same size gets changed in Y-scale. In this changed picture, do all objects still have the exact same size?
Thank you very much!
As long as the change in Y-scale is applied the same to all 3 bars then the area will be the same; however, the perimeter of each bar will be different. As an example, for figure A - using a 6" x 1" bar as an example and multiplying the Y-scale by 1.5. See attached image for all of the dimensions, areas, and perimeters
Let A=(a1, a2) and B=(b1, b2) are vectors forming two neighbor sides of a parallelogram ( rectangular in the question). Its square is
S = | A x B | = | a1 b2 - a2 b1 |.
The transformation
x1 = k1 x
y1 = k2 y
translates A and B to
A1 = (k1 a1, k2 a2),
B1= (k1 b1, k2 b2).
(In the question k1=1).
New square
S1 = | A1 x B1 | = | k1 a1 k2 b2 - k1 a2 k2 b1 | = | k1 k2 | S.
Thus, all initially equal squares remain equal.
By the way, this result remains true for regions of the arbitrary shape with the equal squares, because they can be triangulated.
Dear Mathias Roth,
In addition to the helpful answers, I have a little bit to add:
The literature is full of different formulas that work
for the enlargement called dilation with a scale factor.
It works for geometrical figures in 2D or 3D dimensions.
See for example:
(1) https://www.youtube.com/watch?v=BCllaARDOWI
(2) https://mathbitsnotebook.com/Algebra1/.../FNGTransformationDilation.ht
(3)https://www.khanacademy.org/...transformations...dilations/.../thinkin..
Best regards
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