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I am creating a golden medal (and later silver and bronze) and I want to deboss and emboss the logo into the medal. I have already tried a view things, like having multiple fills, transform them to opposite positions and blur them. As well as trying the extrude effect from Illustrator, but this only works okayish if there is an angle, which I don't have.

I looked at these:

I want to achieve something like this:
enter image description here

And this is the closest I could get, but it just isn't as crisp as my example:
enter image description here

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You should use Illustrator's 3D effect Extrude & Bevel. Your on blur based version cannot be sharp. Beveling causes often harmful artifacts (=self-cutting surfaces and other math difficulties) with letters which have sharp corners. But there's a workaround to get both shadows and highlights without bevel.

You are right, it needs a tilted viewing angle, but the vertical squeezing caused by the angle will be compensated. You get this:

enter image description here

Start by stacking your metal plate gradient and the logo filled with 50% grey RGB(128. 128, 128). Insert to the logo shape 3D filter Extrude & Bevel with only 2 degree tilting:

enter image description here

The black shading is visible with 2 degree pitch angle if there's long enough extrude. The ambient light is turned down to get deep shadow. It can be adjusted by changing the shading color or by inserting some ambient light. The light is nearly straight on the face to get the original fill color, not black. It can be turned a little aside to get some shadow to exactly vertical long lines.

You want also highlight. Make a copy of the extruded logo (=paste in place) and reopen its Extrude & Bevel dialog by clicking the effect in the appearance panel. Change the tilting angle to -2 degrees and change the shading color to nearly white yellow to get the highlight. See NOTE1:

enter image description here

Select both logo shapes and give to them blending mode Hard Light in the Transparency Panel. The underlaying metal plate color jumps visible:

enter image description here

NOTE1: With blending mode Hard Light 50% grey causes nothing. Turning the light a little aside changes the place of the highlight so that the long vertical edges become visible and a little smaller brightness than 100% makes the whole logo body visible.

Not asked: A polished metal plate reflects environment and lights. They define the apparent brightness and color changes. Simple gradient can be too simple. In addition there can be some surface imperfectness - at least outside the logo area if we think only the logo is especially well polished. An example:

enter image description here

I painted more stuff to your gradient and made it darker to get relatively brighter logo highlights. The grains are taken from another case, from this: How to apply this film grain effect? The plate is now a bitmap image. In that case also the logo could be embossed in Photoshop with layer style Bevel&Emboss, which is much faster to use than Illustrator's doubled extrude.

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