When I asked you if the diagram showing the sun in the center of the orbit is violating Kepler's first law, you answered that it is not, because Earth's orbit is nearly a " perfect circle" because the focus is very close to the orbit center, and when I responded that the line matching the two Equinoxes (sunlight is orthogonal to the equator at both Equinoxes) will devise the orbit into two unequal halves then the spaces of the four quarters of the orbit space are not equal, you came with your equations to prove that the four quarters, due to movement, must be equal, because stable diagram can't tell all the truth, although the stable image in our case shows the spaces that to be swept during movement, and that shouldn't transform the big space to a little one or the little to a bigger, exactly like a street of 3 km long, when you move through it, it will never be 5km or 2 km.
I wanted to say: if you don't see any problem to approximate everything concerning elliptical orbits to comply with circular orbits of Copernicus, would you tell me what is the change that Kepler mad by his first law that we can consider as an addition to Copernican law of circular orbits?
In the case of Copernican circular orbit we can say also that the straight line between the planet and the sun sweeps equal areas in equal intervals of time, but in this case we must remember that the planet should move with a uniform velocity where is no acceleration or deceleration, and this conduct us to ask this question: since you insist that the four quarters of orbit space are equal, do you agree that the Earth revolves around the Sun with a uniform velocity? means, without acceleration or deceleration, neither perihelion or aphelion?
Remember, if you don't agree, this will be an evidence that the four quarters are not equal in their spaces which should mean that: the straight line between the planet and the Sun sweeps "unequal" areas in equal intervals of time, which mean the end of Kepler second law.
Joining the above to your strange way of answering my question about the law of equal areas, where you made me feel as if I'm against Kepler and you are defending him from imagined attack by my questions, and this not true, I have a great respect for Kepler but in the same time my respect to truth is much more greater.
"since you insist that the four quarters of orbit space are equal, do you agree that the Earth revolves around the Sun with a uniform velocity? means, without acceleration or deceleration, neither perihelion or aphelion?"
Of course not! Again, this shows you fail to grasp the physics of the situation and the basis for Kepler's 2nd law (equal areas in equal intervals of time)
Obviously and clearly from the physics the velocity of Earth cannot be the same or be uniform through the whole orbit. It must speed up at perihelion and slow down at aphelion. THAT is the point! That is why the equal areas are different in shape at those points!
Study again the shapes of the areas at aphelion and perihelion (as well as at other points) in this animation:
Note how the area is narrower (as a sector of the ellipse) at greater distance and wider (greater angle theta at lesser distance). Hence, it makes sense near perihelion (TOP of image) the area sector (e.g. angle) is wider and at aphelion (BOTTOM of image) the area sector is narrower.
BUT the radius vectors bounding the angles are shorter for case 1 and longer for case 2. Hence, in the end the areas are EQUAL.
Lastly, to be clear, I did NOT say I "saw no problem" in approximating elliptical orbits with circular ones. What I said is that different orbital diagrams have different emphases and must be interpreted in the context of the particular emphasis. Thus, showing the seasons - where the emphasis is on axial tilt- then the elliptical nature of the orbit would be secondary, hence ignored for all practical purposes. By contrast, showing the law of equal areas the emphasis will be on the elliptical shape of the orbit and axial tilt disregarded as well as equinoxes, solstices. But for some reason you insist on conflating the two types of diagrams to try to establish some "truth" apparent only to you.
Again, I suggest going to another expert as I feel this exchange has run its course. I have tried my best to give you the basis for the 2nd law's validity which is well established - so it's not a matter of "truth". Of course it is valid and proven because our observations disclose it to be so.
Anyway, be advised no further questions will be addressed on this topic.