Struct CubicSegment

pub struct CubicSegment<P: VectorSpace> {
    pub coeff: [P; 4],
}

A segment of a cubic curve, used to hold precomputed coefficients for fast interpolation. It is a Curve with domain [0, 1].

Segments can be chained together to form a longer compound curve.

Fields

coeff: [P; 4]

Polynomial coefficients for the segment.

Implementations

impl<P: VectorSpace> CubicSegment<P>

pub fn position(&self, t: f32) -> P

Instantaneous position of a point at parametric value t.

pub fn velocity(&self, t: f32) -> P

Instantaneous velocity of a point at parametric value t.

pub fn acceleration(&self, t: f32) -> P

Instantaneous acceleration of a point at parametric value t.

impl CubicSegment<Vec2>

The CubicSegment<Vec2> can be used as a 2-dimensional easing curve for animation.

The x-axis of the curve is time, and the y-axis is the output value. This struct provides methods for extremely fast solves for y given x.

pub fn new_bezier(p1: impl Into<Vec2>, p2: impl Into<Vec2>) -> Self

Construct a cubic Bezier curve for animation easing, with control points p1 and p2. A cubic Bezier easing curve has control point p0 at (0, 0) and p3 at (1, 1), leaving only p1 and p2 as the remaining degrees of freedom. The first and last control points are fixed to ensure the animation begins at 0, and ends at 1.

This is a very common tool for UI animations that accelerate and decelerate smoothly. For example, the ubiquitous “ease-in-out” is defined as (0.25, 0.1), (0.25, 1.0).

pub fn ease(&self, time: f32) -> f32

Given a time within 0..=1, returns an eased value that follows the cubic curve instead of a straight line. This eased result may be outside the range 0..=1, however it will always start at 0 and end at 1: ease(0) = 0 and ease(1) = 1.

let cubic_bezier = CubicSegment::new_bezier((0.25, 0.1), (0.25, 1.0));
assert_eq!(cubic_bezier.ease(0.0), 0.0);
assert_eq!(cubic_bezier.ease(1.0), 1.0);

How cubic easing works

Easing is generally accomplished with the help of “shaping functions”. These are curves that start at (0,0) and end at (1,1). The x-axis of this plot is the current time of the animation, from 0 to 1. The y-axis is how far along the animation is, also from 0 to 1. You can imagine that if the shaping function is a straight line, there is a 1:1 mapping between the time and how far along your animation is. If the time = 0.5, the animation is halfway through. This is known as linear interpolation, and results in objects animating with a constant velocity, and no smooth acceleration or deceleration at the start or end.

y
│         ●
│       ⬈
│     ⬈    
│   ⬈
│ ⬈
●─────────── x (time)

Using cubic Beziers, we have a curve that starts at (0,0), ends at (1,1), and follows a path determined by the two remaining control points (handles). These handles allow us to define a smooth curve. As time (x-axis) progresses, we now follow the curve, and use the y value to determine how far along the animation is.

y
         ⬈➔●
│      ⬈   
│     ↑      
│     ↑
│    ⬈
●➔⬈───────── x (time)

To accomplish this, we need to be able to find the position y on a curve, given the x value. Cubic curves are implicit parametric functions like B(t) = (x,y). To find y, we first solve for t that corresponds to the given x (time). We use the Newton-Raphson root-finding method to quickly find a value of t that is very near the desired value of x. Once we have this we can easily plug that t into our curve’s position function, to find the y component, which is how far along our animation should be. In other words:

Given time in 0..=1

Use Newton’s method to find a value of t that results in B(t) = (x,y) where x == time

Once a solution is found, use the resulting y value as the final result

Trait Implementations

impl<P: Clone + VectorSpace> Clone for CubicSegment<P>

fn clone(&self) -> CubicSegment<P>

Returns a copy of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl<P: VectorSpace> Curve<P> for CubicSegment<P>

Available on crate feature curve only.

fn domain(&self) -> Interval

The interval over which this curve is parametrized.

fn sample_unchecked(&self, t: f32) -> P

Sample a point on this curve at the parameter value t, extracting the associated value. This is the unchecked version of sampling, which should only be used if the sample time t is already known to lie within the curve’s domain.

fn sample(&self, t: f32) -> Option<T>

Sample a point on this curve at the parameter value t, returning None if the point is outside of the curve’s domain.

fn sample_clamped(&self, t: f32) -> T

Sample a point on this curve at the parameter value t, clamping t to lie inside the domain of the curve.

fn sample_iter( &self, iter: impl IntoIterator<Item = f32>, ) -> impl Iterator<Item = Option<T>>

where Self: Sized,

Sample a collection of n >= 0 points on this curve at the parameter values t_n, returning None if the point is outside of the curve’s domain.

fn sample_iter_unchecked( &self, iter: impl IntoIterator<Item = f32>, ) -> impl Iterator<Item = T>

where Self: Sized,

Sample a collection of n >= 0 points on this curve at the parameter values t_n, extracting the associated values. This is the unchecked version of sampling, which should only be used if the sample times t_n are already known to lie within the curve’s domain.

fn sample_iter_clamped( &self, iter: impl IntoIterator<Item = f32>, ) -> impl Iterator<Item = T>

where Self: Sized,

Sample a collection of n >= 0 points on this curve at the parameter values t_n, clamping t_n to lie inside the domain of the curve.

fn map<S, F>(self, f: F) -> MapCurve<T, S, Self, F>

where Self: Sized, F: Fn(T) -> S,

Create a new curve by mapping the values of this curve via a function f; i.e., if the sample at time t for this curve is x, the value at time t on the new curve will be f(x).

fn reparametrize<F>(self, domain: Interval, f: F) -> ReparamCurve<T, Self, F>

where Self: Sized, F: Fn(f32) -> f32,

Create a new Curve whose parameter space is related to the parameter space of this curve by f. For each time t, the sample from the new curve at time t is the sample from this curve at time f(t). The given domain will be the domain of the new curve. The function f is expected to take domain into self.domain().

fn reparametrize_linear( self, domain: Interval, ) -> Result<LinearReparamCurve<T, Self>, LinearReparamError>

where Self: Sized,

Linearly reparametrize this Curve, producing a new curve whose domain is the given domain instead of the current one. This operation is only valid for curves with bounded domains; if either this curve’s domain or the given domain is unbounded, an error is returned.

fn reparametrize_by_curve<C>(self, other: C) -> CurveReparamCurve<T, Self, C>

where Self: Sized, C: Curve<f32>,

Reparametrize this Curve by sampling from another curve.

fn graph(self) -> GraphCurve<T, Self>

where Self: Sized,

Create a new Curve which is the graph of this one; that is, its output echoes the sample time as part of a tuple.

fn zip<S, C>( self, other: C, ) -> Result<ZipCurve<T, S, Self, C>, InvalidIntervalError>

where Self: Sized, C: Curve<S> + Sized,

Create a new Curve by zipping this curve together with another.

fn chain<C>(self, other: C) -> Result<ChainCurve<T, Self, C>, ChainError>

where Self: Sized, C: Curve<T>,

Create a new Curve by composing this curve end-to-start with another, producing another curve with outputs of the same type. The domain of the other curve is translated so that its start coincides with where this curve ends.

fn reverse(self) -> Result<ReverseCurve<T, Self>, ReverseError>

where Self: Sized,

Create a new Curve inverting this curve on the x-axis, producing another curve with outputs of the same type, effectively playing backwards starting at self.domain().end() and transitioning over to self.domain().start(). The domain of the new curve is still the same.

fn repeat(self, count: usize) -> Result<RepeatCurve<T, Self>, RepeatError>

where Self: Sized,

Create a new Curve repeating this curve N times, producing another curve with outputs of the same type. The domain of the new curve will be bigger by a factor of n + 1.

fn forever(self) -> Result<ForeverCurve<T, Self>, RepeatError>

where Self: Sized,

Create a new Curve repeating this curve forever, producing another curve with outputs of the same type. The domain of the new curve will be unbounded.

fn ping_pong(self) -> Result<PingPongCurve<T, Self>, PingPongError>

where Self: Sized,

Create a new Curve chaining the original curve with its inverse, producing another curve with outputs of the same type. The domain of the new curve will be twice as long. The transition point is guaranteed to not make any jumps.

fn chain_continue<C>( self, other: C, ) -> Result<ContinuationCurve<T, Self, C>, ChainError>

where Self: Sized, T: VectorSpace, C: Curve<T>,

Create a new Curve by composing this curve end-to-start with another, producing another curve with outputs of the same type. The domain of the other curve is translated so that its start coincides with where this curve ends.

fn resample<I>( &self, segments: usize, interpolation: I, ) -> Result<SampleCurve<T, I>, ResamplingError>

where Self: Sized, I: Fn(&T, &T, f32) -> T,

Resample this Curve to produce a new one that is defined by interpolation over equally spaced sample values, using the provided interpolation to interpolate between adjacent samples. The curve is interpolated on segments segments between samples. For example, if segments is 1, only the start and end points of the curve are used as samples; if segments is 2, a sample at the midpoint is taken as well, and so on. If segments is zero, or if this curve has an unbounded domain, then a ResamplingError is returned.

fn resample_auto( &self, segments: usize, ) -> Result<SampleAutoCurve<T>, ResamplingError>

where Self: Sized, T: StableInterpolate,

Resample this Curve to produce a new one that is defined by interpolation over equally spaced sample values, using automatic interpolation to interpolate between adjacent samples. The curve is interpolated on segments segments between samples. For example, if segments is 1, only the start and end points of the curve are used as samples; if segments is 2, a sample at the midpoint is taken as well, and so on. If segments is zero, or if this curve has an unbounded domain, then a ResamplingError is returned.

fn samples( &self, samples: usize, ) -> Result<impl Iterator<Item = T>, ResamplingError>

where Self: Sized,

Extract an iterator over evenly-spaced samples from this curve. If samples is less than 2 or if this curve has unbounded domain, then an error is returned instead.

fn resample_uneven<I>( &self, sample_times: impl IntoIterator<Item = f32>, interpolation: I, ) -> Result<UnevenSampleCurve<T, I>, ResamplingError>

where Self: Sized, I: Fn(&T, &T, f32) -> T,

Resample this Curve to produce a new one that is defined by interpolation over samples taken at a given set of times. The given interpolation is used to interpolate adjacent samples, and the sample_times are expected to contain at least two valid times within the curve’s domain interval.

fn resample_uneven_auto( &self, sample_times: impl IntoIterator<Item = f32>, ) -> Result<UnevenSampleAutoCurve<T>, ResamplingError>

where Self: Sized, T: StableInterpolate,

Resample this Curve to produce a new one that is defined by automatic interpolation over samples taken at the given set of times. The given sample_times are expected to contain at least two valid times within the curve’s domain interval.

fn by_ref(&self) -> &Self

where Self: Sized,

Borrow this curve rather than taking ownership of it. This is essentially an alias for a prefix &; the point is that intermediate operations can be performed while retaining access to the original curve.

fn flip<U, V>(self) -> impl Curve<(V, U)>

where Self: Sized + Curve<(U, V)>,

Flip this curve so that its tuple output is arranged the other way.

impl<P: Debug + VectorSpace> Debug for CubicSegment<P>

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl<P: Default + VectorSpace> Default for CubicSegment<P>

fn default() -> CubicSegment<P>

Returns the “default value” for a type.

impl<'de, P> Deserialize<'de> for CubicSegment<P>

where P: Deserialize<'de> + VectorSpace,

fn deserialize<__D>(__deserializer: __D) -> Result<Self, __D::Error>

where __D: Deserializer<'de>,

Deserialize this value from the given Serde deserializer.

impl<P: VectorSpace> Extend<CubicSegment<P>> for CubicCurve<P>

fn extend<T: IntoIterator<Item = CubicSegment<P>>>(&mut self, iter: T)

Extends a collection with the contents of an iterator.

fn extend_one(&mut self, item: A)

🔬This is a nightly-only experimental API. (extend_one)

Extends a collection with exactly one element.

fn extend_reserve(&mut self, additional: usize)

🔬This is a nightly-only experimental API. (extend_one)

Reserves capacity in a collection for the given number of additional elements.

impl<P: VectorSpace> From<CubicSegment<P>> for RationalSegment<P>

fn from(value: CubicSegment<P>) -> Self

Converts to this type from the input type.

impl<P> FromReflect for CubicSegment<P>

where CubicSegment<P>: Any + Send + Sync, P: TypePath + VectorSpace, [P; 4]: FromReflect + TypePath + MaybeTyped + RegisterForReflection,

fn from_reflect(reflect: &dyn PartialReflect) -> Option<Self>

Constructs a concrete instance of Self from a reflected value.

fn take_from_reflect( reflect: Box<dyn PartialReflect>, ) -> Result<Self, Box<dyn PartialReflect>>

Attempts to downcast the given value to Self using, constructing the value using from_reflect if that fails.

impl<P> GetTypeRegistration for CubicSegment<P>

where CubicSegment<P>: Any + Send + Sync, P: TypePath + VectorSpace, [P; 4]: FromReflect + TypePath + MaybeTyped + RegisterForReflection,

fn get_type_registration() -> TypeRegistration

Returns the default TypeRegistration for this type.

fn register_type_dependencies(registry: &mut TypeRegistry)

Registers other types needed by this type.

impl<P: PartialEq + VectorSpace> PartialEq for CubicSegment<P>

fn eq(&self, other: &CubicSegment<P>) -> bool

Tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl<P> PartialReflect for CubicSegment<P>

where CubicSegment<P>: Any + Send + Sync, P: TypePath + VectorSpace, [P; 4]: FromReflect + TypePath + MaybeTyped + RegisterForReflection,

fn get_represented_type_info(&self) -> Option<&'static TypeInfo>

Returns the TypeInfo of the type represented by this value.

fn clone_value(&self) -> Box<dyn PartialReflect>

Clones the value as a Reflect trait object.

fn try_apply(&mut self, value: &dyn PartialReflect) -> Result<(), ApplyError>

Tries to apply a reflected value to this value.

fn reflect_kind(&self) -> ReflectKind

Returns a zero-sized enumeration of “kinds” of type.

fn reflect_ref(&self) -> ReflectRef<'_>

Returns an immutable enumeration of “kinds” of type.

fn reflect_mut(&mut self) -> ReflectMut<'_>

Returns a mutable enumeration of “kinds” of type.

fn reflect_owned(self: Box<Self>) -> ReflectOwned

Returns an owned enumeration of “kinds” of type.

fn try_into_reflect( self: Box<Self>, ) -> Result<Box<dyn Reflect>, Box<dyn PartialReflect>>

Attempts to cast this type to a boxed, fully-reflected value.

fn try_as_reflect(&self) -> Option<&dyn Reflect>

Attempts to cast this type to a fully-reflected value.

fn try_as_reflect_mut(&mut self) -> Option<&mut dyn Reflect>

Attempts to cast this type to a mutable, fully-reflected value.

fn into_partial_reflect(self: Box<Self>) -> Box<dyn PartialReflect>

Casts this type to a boxed, reflected value.

fn as_partial_reflect(&self) -> &dyn PartialReflect

Casts this type to a reflected value.

fn as_partial_reflect_mut(&mut self) -> &mut dyn PartialReflect

Casts this type to a mutable, reflected value.

fn reflect_partial_eq(&self, value: &dyn PartialReflect) -> Option<bool>

Returns a “partial equality” comparison result.

fn debug(&self, f: &mut Formatter<'_>) -> Result

Debug formatter for the value.

fn apply(&mut self, value: &(dyn PartialReflect + 'static))

Applies a reflected value to this value.

fn reflect_hash(&self) -> Option<u64>

Returns a hash of the value (which includes the type).

fn serializable(&self) -> Option<Serializable<'_>>

Returns a serializable version of the value.

fn is_dynamic(&self) -> bool

Indicates whether or not this type is a dynamic type.

impl<P> Reflect for CubicSegment<P>

where CubicSegment<P>: Any + Send + Sync, P: TypePath + VectorSpace, [P; 4]: FromReflect + TypePath + MaybeTyped + RegisterForReflection,

fn into_any(self: Box<Self>) -> Box<dyn Any>

Returns the value as a Box<dyn Any>.

fn as_any(&self) -> &dyn Any

Returns the value as a &dyn Any.

fn as_any_mut(&mut self) -> &mut dyn Any

Returns the value as a &mut dyn Any.

fn into_reflect(self: Box<Self>) -> Box<dyn Reflect>

Casts this type to a boxed, fully-reflected value.

fn as_reflect(&self) -> &dyn Reflect

Casts this type to a fully-reflected value.

fn as_reflect_mut(&mut self) -> &mut dyn Reflect

Casts this type to a mutable, fully-reflected value.

fn set(&mut self, value: Box<dyn Reflect>) -> Result<(), Box<dyn Reflect>>

Performs a type-checked assignment of a reflected value to this value.

impl<P> Serialize for CubicSegment<P>

where P: Serialize + VectorSpace,

fn serialize<__S>(&self, __serializer: __S) -> Result<__S::Ok, __S::Error>

where __S: Serializer,

Serialize this value into the given Serde serializer.

impl<P> Struct for CubicSegment<P>

where CubicSegment<P>: Any + Send + Sync, P: TypePath + VectorSpace, [P; 4]: FromReflect + TypePath + MaybeTyped + RegisterForReflection,

fn field(&self, name: &str) -> Option<&dyn PartialReflect>

Returns a reference to the value of the field named name as a &dyn PartialReflect.

fn field_mut(&mut self, name: &str) -> Option<&mut dyn PartialReflect>

Returns a mutable reference to the value of the field named name as a &mut dyn PartialReflect.

fn field_at(&self, index: usize) -> Option<&dyn PartialReflect>

Returns a reference to the value of the field with index index as a &dyn PartialReflect.

fn field_at_mut(&mut self, index: usize) -> Option<&mut dyn PartialReflect>

Returns a mutable reference to the value of the field with index index as a &mut dyn PartialReflect.

fn name_at(&self, index: usize) -> Option<&str>

Returns the name of the field with index index.

fn field_len(&self) -> usize

Returns the number of fields in the struct.

fn iter_fields(&self) -> FieldIter<'_>

Returns an iterator over the values of the reflectable fields for this struct.

fn clone_dynamic(&self) -> DynamicStruct

Clones the struct into a DynamicStruct.

fn get_represented_struct_info(&self) -> Option<&'static StructInfo>

Will return None if TypeInfo is not available.

impl<P> TypePath for CubicSegment<P>

where CubicSegment<P>: Any + Send + Sync, P: TypePath + VectorSpace,

fn type_path() -> &'static str

Returns the fully qualified path of the underlying type.

fn short_type_path() -> &'static str

Returns a short, pretty-print enabled path to the type.

fn type_ident() -> Option<&'static str>

Returns the name of the type, or None if it is anonymous.

fn crate_name() -> Option<&'static str>

Returns the name of the crate the type is in, or None if it is anonymous.

fn module_path() -> Option<&'static str>

Returns the path to the module the type is in, or None if it is anonymous.

impl<P> Typed for CubicSegment<P>

where CubicSegment<P>: Any + Send + Sync, P: TypePath + VectorSpace, [P; 4]: FromReflect + TypePath + MaybeTyped + RegisterForReflection,

fn type_info() -> &'static TypeInfo

Returns the compile-time info for the underlying type.

impl<P: Copy + VectorSpace> Copy for CubicSegment<P>

impl<P: VectorSpace> StructuralPartialEq for CubicSegment<P>