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Jake Blocker's excellent answer explains what a dogleg maneuver is, what it's for, what it looks like, and even what a real dog's leg looks like for comparison!

A dogleg maneuver is done to change the inclination of a certain payload, and the reason it limits the payload capacity is most likely due to the cosine losses.

In this answer I mention an incident which might be the reason that dogleging to orbit was invented, but I don't know.

Question: Which launch was the first to dogleg?

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    $\begingroup$ ...intentionally? ;) $\endgroup$ – SF. Jun 5 '18 at 14:21
  • $\begingroup$ @SF. ha! Yes let's say a planned trajectory $\endgroup$ – uhoh Jun 5 '18 at 14:25
  • $\begingroup$ Could this be it? en.wikipedia.org/wiki/Thor-Ablestar $\endgroup$ – uhoh Jun 6 '18 at 1:22
  • $\begingroup$ After the bounty expires as well as its 24 hour grace period I'm going to post that link as an answer, unless someone beats me to it (and therefore collects the bounty as well!) $\endgroup$ – uhoh Sep 8 '18 at 12:36
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    $\begingroup$ I think, ISRO mostly uses dogleg maneuver $\endgroup$ – Amar Sep 8 '18 at 13:59

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