Is there any specific wand movement for the Patronus Charm?

Question

We know that some charms need exact hand movement like Wingardium Leviosa. Is there such specific wand movement for Patronus Charm too? I have visited the Harry Potter Wikia of Patronus Charm. It says there is a specific wand movement. Below is the image.

This hand movement is used by Luna Lovegood and some others while practicing the charm when they were part of Dumbledore's Army. Also in the movie Harry Potter and the Deathly Hallows part 2, Severus Snape uses this specific hand movement to show his Doe Patronus to Dumbledore.

But while conjuring his first corporeal Patronus, Harry just points his wand into air and shouts Expecto Patronum. The spell works and his stag Patronus fends off 100+ dementors.

So, Is there any specific wand movement for Expecto Patronum?

Answer 1

No.

When Lupin teaches Harry to do the spell there is no mention of hand movement at all.

But we know that some wizards can do magic even without words or wands.

I would generally guess that also movements are not necessarily a part of the spell itself. It is only something that helps you to do the spell.

To cite Pottermore: The majority of witches and wizards are unable to produce Patronuses and to do so is generally considered a mark of superior magical ability.

So, as Patronus charm is very difficult it is possible, that there is some movement that someone made up to make this spell easier. But it is not mentioned in the canon.