To begin broadly, the relationships between Prospero and Miranda, Ariel and Caliban seem linked in the sense that Prospero tends to continually express his control over each - whether it be subtly (as with Miranda), explicitly (Caliban) or a mixture of the two (Ariel).
Consider the first dialogue between Prospero and Miranda: after consoling her in regards to the storm he has helped produce, he informs Miranda of their journey to the island together, occasionally emphasizing the fact that he alone kept her safe and raised her, referring to himself even as “thy schoolmaster” who “made thee more profit than other princes can” (Pg. 12, line 172). He doesn’t present his authority in an off-putting manner – his love for Miranda as his daughter seems genuine – but he portions it out in gentle waves nonetheless.
It is the opposite case with Caliban. From his entrance onward, Caliban receives the brunt of Prospero’s control. With each subsequent threat (“tonight thou shalt have cramps” - Pg. 18, line 325) and coarse word, as well as the simple fact that Prospero has confined Caliban to a rock, it becomes more apparent that Prospero finds no limit when expressing his power (literal and otherwise) over Caliban.
As for his relationship with Ariel, one finds that Prospero utilizes both ends of the spectrum in signifying his control over him. While Prospero seems to understand that Ariel’s actions have been exceedingly helpful to him, commending his efforts (“Ariel, thy charge exactly as performed” – Pg. 15, line 236) and so forth, he wastes no time in violently reminding Ariel of their “business arrangement.” By asking “Dost thou forget from what torment I did free thee?” (Pg. 15, line 250) Prospero emphasizes Ariel’s enormous debt to him, further asserting his weighty control over him.
