Yes, much of the terminology we use with dogs comes from horses. We generally refer to the walk, trot and gallop just like a horse. Dogs do also have a canter, or lope.
They will also "pace" or amble with the legs on one side alternating movement just like a horse. In dogs, the pace is sometimes an adaptation to some structural problem or a tired dog may pace in order to rest the hard working trotting muscles. In some breeds , like the Old English Sheepdog, the pace, or amble is a natural gait.
Dogs will even change leads, although the process is slightly different than in a horse. People who run agility learn to be aware of lead changes to help the dog negotiate jumps with less wear and tear on the dog. As a four-footed animal, the dog moves much the same.
Dogs are judged in shows at a trot, because dogs are natural trotting animals, who like the wolf, should be able to cover a lot of ground at a trot without tiring. Sight hounds, of course are bred to run and run fast. You can see it in how differently they are built from the other breeds.
Although often overlooked in a modern dog, movement is dependant on good structure and conditioning and can impact health and soundness especially in dogs who still hunt, herd, guard,pull sleds or compete in dog sports. Compare the heavy powerful "draft" build of a weight-pulling Malamute to the lean and quick Siberian Husky. Some of the toy breeds are even required to have a "hackney" gait, imitating the Hackney horse breed.
Since agility started getting popular, it seems that handlers are wising up to good structure, and the staying power that it gives to their dogs, both in competition and into old age. This is already creating a demand among the show set for better structure. If you get a book on dog breeds that has the breed standards, you will see familiar horse terms in many of them, especially in breeds bred to work.
So...yes, much in common with horses! Good question, fun to answer!
Yeah! Sure they can :) Usually the more active ones walk/run that way. My dog trots :P.
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