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I had this image done some time ago in Adobe Illustrator and I saved it in .pdf because I need to put it into a .tex (LaTeX) file. The pdf file is avaliable in https://ufile.io/3tol20xu.

enter image description here

How I can improve the hemisphere to look similar to this 3D figure?

enter image description here

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The question has seemingly been shortened to contain a 3D appearance construction problem. Let's start by removing the unnecessary blue curves and by checking how consistent drawing you actually have:

enter image description here

Y and Z axles are stretched to cover the bottom circle area and the visible outline curve of the expected sphere is drawn

The projection is widely used in math illustrations, but geometrically it's distorted in a tricky way and causes easily contradictions. The XZ plane half circle is exactly circular. The viewing direction is perpendicular to XZ plane. But the ZY plane green circle is not seen sideways. The camera is not in infinity, it's in a place where X, Y and Z all are positive and the view direction is straight towards the XZ plane.

There should be a perspective. But the pink cutting plane circle is drawn by scaling the green one proportionally. That trick is right only for a parallel projection. The result: The pink one cannot meet right the spherical outline and the blue arcs.

As a quick partial fix I tried to scale it non-proportionally to make it meet the blue arcs and the outline circle better:

enter image description here

It's only eyeballed and closer inspection shows that it's proportions are still OFF. The X axis punches it at a little wrong place. The punching point should be at the crossing of the main axles of the ellipse, but it isn't.

The quick fix isn't enough because the right outline of the sphere should be elliptical in a perspective image where the camera doesn't point towards the sphere. That error in mine.

Assuming your drawing is perfect and sketching the outline of the sphere around it gives this (rough) result.

enter image description here

It may be the right perspective image of a sphere, but it looks non-spherical. Both versions look so wrong that I wouldn't use it as a math illustration. I would use a 3D program for the geometry construction or I would draw it as an isometric 2D drawing. They can both be done to wanted dimensions without eyeballing - only known numbers or snapping to make all fit. Here's an example of such isometric 2D version:

enter image description here

But that was not the main point in the question. The main point was how to construct the partially transparent half sphere dome. It's done also in the shown example.

Start by duplicating the sphere outline circle and the bottom circle in the XZ plane.

enter image description here

A) The duplicates are moved apart.

B) Select both, combine with the shape builder what's needed for the dome. Delete all remnants - there's sometimes extra pieces left at the edges and there can be groups if the originals had groups.

C) Change the fill color, if needed

D) Insert effect Inner Glow. Use a dark, but saturated color. The effect can be reopened in the appearance panel later for adjustments.

Finally reduce the opacity and move the dome on the image. Remove the outline circle and make the bottom circle less saturated.

enter image description here

As said above: the starting point was too difficult for only averagely skilled person like me. The result is extremely distorted. To get it right I recommend scrapping the PDF and restarting in a 3D program or making it as an isometric drawing.

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  • Excuse me for the delay. I am not able to generate your work. I appreciate very much every your efforts. The red circle (red section) seem that is not parallel with the geen section. +1.
    – Sebastiano
    Commented May 11 at 21:26
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    @Sebastiano Your projection in PDF is difficult. It needs a skilled artist. To make it look right the red disk should have totally different shape than the green one due the perspective. Your projection is NOT a parallel projection. The fact that the viewing direction is not towards to the green nor the red disk , but much towards the right makes things even more difficult. Your image is like a slice cut from the left edge of a big photo shot with a wide view camera. Draw an isometric version to get it right easily as a 2D drawing. Or build it all in a 3D program at least as a wireframe. Commented May 11 at 21:45
  • I have few skills in this field.
    – Sebastiano
    Commented May 11 at 21:52
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    @Sebastiano Isometric images can be constructed with an isometric grid. Isometric mages of circles can be created by rounding isometric squares. XY, XZ and TZ plane isometric shapes can be made also by skewing, rotating and scaling with well known fixed numbers. Numerous tutorials exist. And you can also ask something. Commented May 11 at 22:00
  • I am too busy in this period. :-(((( I am not able to read documentation. Thank you very much for your work.
    – Sebastiano
    Commented May 11 at 22:02

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