Can two figures have the same area, perimeter, and same number of segments have different shape? I want to make an algorithm grouping all the details having the same shape.
each detail is defined by its surface, and a list of contour lines.
First I believed that having the same perimeter length and same surface would be enough, but I saw on that link that it is wrong hypothesis.
If I take as additional condition that the two shapes have the same number of segments, would it be enough? Or else how can I check that?
The problem is for each detail, they can get rotation, or symmetry.
Edit :
Thanks for your answers, I finally found a way to solve the problem (answer below)
 A: In the spirit of the no-words answer to the linked question:

A: 
Possibly the simplest counterexample: Form two sides of a triangle with line segments of unequal length. In one version, mirror it. In the other, rotate it 180°.
A: Inspired by Hagen von Eitzen's answer, a particular pair of tetrominoes furnishes another counterexample (minimal among polyominoes):

A: Well, I finally managed to do that, but it was longer than expected :
My shape is defined by a list of Contour element, Contour is defined by 2 points and a radius.If this is a segment then radius=0, if this is an arc, radius is positive if I turn in trigonometric direction, negative if opposite side).
I make a first check, checking if areas are equal(just to identify faster if shapes are equal or not).
For each shape, I browse each segment(or circle arc) ClockWise direction and I return 3 results :


*

*List of the lengths of segments(just Pythagoras, don't check radiuses)

*List of radiuses of each segment/arc

*List of angles between each segment and consecutive segment


I then can compare them (let's take in account that I start from the same point on each figure). In C# I just made a loop trying to start from different points.
If all 3 lists are equals, this means the shapes are equal and without rotation.
Then for symmetry problem I just will recalculate 3 above lists for second shape, but browsing contours in opposite direction, if the results are equal(angles and radiuses just will be opposite sign), so it is the same shape with symmetry.
A: No, it is still not enough.  Even for quadrilaterals it is not enough.  Take a kite with sides $1,1,3,3$ and the angle between the two $1$s a right angle.  The perimeter is $8$ and the area is $\frac 12(1+\sqrt{17})$.  Now take a rhombus with sides of $8$.  It also has a perimeter of $8$ and you can choose the angle to make the areas match.  It is even worse for more sides.  The area and perimeter are just two constraints, while there are lots of degrees of freedom.
A: Yes, as others have demonstrated, you can have different shapes given the same area, perimeter, and number of segments.
Furthermore, it seems to me you can have different shapes given any number of arbitrary metrics.
Take this terrible looking key, for instance:

You could easily imagine an infinite number of keys like this each uniquely represented by 7 numbers.
For this key, those values would be 2, 3, 1, 2, 3, 1, 2.
These need not be integers.
You could imagine a million-toothed key being uniquely represented with a million numbers.
Now, technically you can use a single real number to represent a million real numbers.
However, I don't believe any such function that does this conversion is continuous, and so likely isn't useful for you.
When they aren't continuous, you'll have metrics like 5.213283 and 5.213284 represent wildly different shapes.
I would consider perimeter and area nice, continuous metrics.
Any small perturbations to the points of the polygon result in a small change in those values.
Note: I could be completely wrong. I look forward to looking up that numberphile video later today.
A: A famous problem was posed in 1966, "Can you hear the shape of a drum?" That is, can you find two different planar domains whose spectrum is the same? It was known that any two such domains must have the same area and the same perimeter. The question was answered negatively by Gordon, Webb and Wolpert in the early 1990s. Here is their example:

Notice that the number of sides is also the same, so this answer's the OP's question with even more conditions imposed. 
