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The original Vostok and Voshkod space vehicles were mostly spherical shapes. The Americans went with vaguely conical/capsule shapes.

But Soyuz from the beginning has had a sort of gumdrop shape for the Descent module. Clearly it works, having only killed 2 crews (?) in many hundreds of launches.

Soyuz Descent module

What were the influences that led to that design choice?

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According to Wikipedia:

One of the design requirements for the reentry module was for it to have the highest possible volumetric efficiency (internal volume divided by hull area). The best shape for this is a sphere, but such a shape can provide no lift, which results in a purely ballistic reentry. Ballistic reentries are hard on the occupants due to high deceleration and cannot be steered beyond their initial deorbit burn. That is why it was decided to go with the "headlight" shape that the Soyuz uses—a hemispherical forward area joined by a barely angled conical section (seven degrees) to a classic spherical section heat shield. This shape allows a small amount of lift to be generated due to the unequal weight distribution. The nickname was thought up at a time when nearly every headlight was circular.

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Phillip Clark and Ralph Gibbons claim that design and configuration was influenced by a proposal from General Electric for a potential Apollo spacecraft in 1961, which was available to the public ("The Evolution of the Soyuz Program", 1983).

Though the orbital module as well as the propulsion module have a different shape than soyuz' you'll see that the descent module looks very familiar: Apollo D-2

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