Calculate bounds of the inner rectangle of the polygon based on it's constant bounds along with border

Question

Yesterday I asked this question seems not clear. So, I am writing new question with mathematical terms only.

Here is some points about shapes I used:

• Square: Rectangle that all angles are 90° & all sides are equal length. So, consider Square also a Rectangle.
• Triangle: I used an equilateral triangle is a triangle in which all three sides have the same length.
• Polygon: A finite number of straight line segments connected to form a closed.

So, Here after I use the term Polygon to describe triangle & rectangle shapes when it's required to mention consider this shape as polygone. Becuase, I used triangle (regular polygon) to simplfy the work to find out the formula.

Variables used in this problem:

• b = Border width

Bounds of Outer Rectangle (r) that enclosing the polygon(p):

• x = Origin x
• y = Origin y
• w = Width
• h = Height

So, If r = (x,y,w,h) then, Bounds of Inner Rectangle that encloses the inner polygon(p1) (r1) = (x1,y1,w1,h1)

Here, inner polygon = Polygon by subtracting the border width.

What is my final goal:

I should able to calculate inner rectangle boundary of any polygon like below image, I have angles, outer rectangle bounds & border width. The problem is to find out the inner rectangle bounds across to the outer rectangle based on the polygon at that enclosed by inner rectangle.

Problem 1:

Consider a Polygon(p) as a shape of rectangle(r) which has width(w) is 100 & height(h) is 100. The border width of the polygon is 10. The what is the inner boundary of the rectangle that encloses the inner polygon?

Note: I solved it by mind calculation(I didn't took angles in the calculation). But we can't apply this same formula for triangle. So, this is not solution for my problem.

Problem 2:

Consider the polygon (don't consider regulard) with three edges and three vertices & all angles are equal in measure.

If the rectangle(r) boundary is zero & size(w,h) is (100,87) that encloses the polygon(p) along with polygon border width as 10, then

What is the bounds of the inner rectangle(r2) that encloses the inner polygon(p2) ?

Hope I clearly explained the problem. If anything I am missed then please let me know. Thanks in advance.

Answer 1

Inner triangle is the image of outer triangle under a homothetic transformation centred at the centre of the triangle.

• The centre of the triangle is located on the altitude, at $1/3$ of its length starting from the base.

• The ratio $r$ of the homothety can be computed as the ratio of the distances from the centre to the midpoint of the bases: $$ r={{1\over3}{\sqrt3\over2}100-10\over {1\over3}{\sqrt3\over2}100}= 1-{\sqrt3\over 5}. $$

Inner rectangle is then the image of outer rectangle under the same homothety.

Answer 2

Let us denote width and height of the inner rectangle as $w_\mathrm{in}$ and $h_\mathrm{in}$ respectively. Then, we have, $$w_\mathrm{in} = w-2\sqrt{3}b. \qquad\mathrm{and}\qquad h_\mathrm{in} = h\space–\space3b.$$

Using these results, we can easily derive the coordinates of the four corners of the sought rectangle $P$, $Q$, $R$, and $S$ as given in the diagram.

Frankly, I am not sure whether these are the results you were looking for. If you have doubts you can always comment on this answer no holds barred.