Overview

When perspectives are switched, the continuous parts of the camera state transition from the state at the moment of the switch to the target state within a fixed duration. Position, rotation, FOV, and orthographic view height may use mutually independent algorithms. Projection mode is a discrete state and directly uses the target value.

Time and progress

Let the transition duration be $D$ and the start time be $t_0$. At time $t$, first compute the normalized time:

$$u=\mathrm{clamp}(\frac{t-t_0}{D},0,1)$$

Then apply the easing function $\varphi$ to obtain the actual progress:

$$p=\mathrm{clamp}(\varphi (u),0,1)$$

The easing function describes how time is allocated across the transition window. It can be a linear function, a smooth curve, or any other function that satisfies the progress constraints; the specific curve only changes how the progress varies over time, not the state formulas of each algorithm. The same $p$ is used by every continuous-field algorithm; therefore they share the fixed transition start and end times but do not have to share the same trajectory.

When $D=0$ or $t-t_0\ge D$, the continuous fields are directly equal to the target state.

Shared goal of the algorithms

Let $x_0$ be the starting state at the moment of the switch and $x_T(t)$ the current target state. If the target does not change during the transition, all algorithms satisfy:

$$x(0)=x_0,x(D)=x_T$$

The target state may also keep changing, for example player movement changes position and mouse input changes orientation.

Continuous and discrete states

Position, rotation, FOV, and orthographic view height are continuous states that can transition smoothly. Projection mode is a discrete choice and cannot be numerically interpolated between perspective and orthographic projection; it becomes the target mode immediately at the switch. Orthographic view height remains a continuous scalar and can therefore vary smoothly on its own.