Therefore if income rises the SLOPE (gradient) of the demand curve changes...to the right.，

Income effect the demand elasticity, because demand of consumer is basis on their purchase ability, for example, if income per capita reduced, they may do not want buy some luxury goods like before. But if the goods is essential ,like food, or drink, there are not a big change in demand. so change in income leading to change the demand, but depends on price elasticity of demand.
Let’s talk about the elastic curve:

1. Perfect elastic:
generally, the good which has the perfect elastic demand which means the demand still change even though the price change or not. So here, I think the demand curve will not change.

Elastic:
Also this depend on the types of goods . For example, when income rise the demand of inferior good may reduce, so the demand curve may shift to the left, and if the goods is luxury goods, the demand may shift to the right, and normal good may expansion.


3. Perfect inelastic:
Not change, because the demand of this goods is inelastic which means does not matter the price change, but I think he size of change in price will effect demand.

4. Inelastic.
Depends on different types of goods, but I think it most likely to contraction, some of then will shift to the right, for example Giffen goods.