9.3
58.15 Graph Helpers and Traced Paths
These helpers construct axes-aware graph annotations and areas from numeric procedures. Tangents are numerical approximations, not symbolic derivatives. traced-path is driven by an explicit phase value and reconstructs its sampled history without depending on previously rendered frames.
See also Function, Parametric, and Data Plots, ODE Trajectories and Streamlines.
58.15.1 Coordinate and Calculus Helpers
Coordinate and calculus helpers construct static, axes-aware teaching diagrams. They evaluate numeric procedures during construction and return ordinary immutable Visuals or points. For an animated construction, place one of these calls inside derived-visual and rebuild it from the sampled parameter value.
procedure
(graph-point axes function x) → vec2?
axes : axes-visual? function : (procedure-arity-includes/c 1) x : finite-real?
Evaluates function at numeric x and converts the resulting
coordinate through axes-coordinates->point. The result is in the
axes’ containing world coordinate system.
The result is a point, not a rendered marker.
Examples:
procedure
(graph-label axes function x label #:id id [ #:offset offset #:font-size font-size #:color color]) → text-visual? axes : axes-visual? function : (procedure-arity-includes/c 1) x : finite-real? label : string? id : symbol? offset : vec2? = (vec2 1/5 1/5) font-size : finite-real? = 1/4 color : any/c = "black"
procedure
(vertical-line-to-graph axes function x #:id id [ #:baseline baseline #:opacity opacity #:stroke stroke #:stroke-width stroke-width]) → path-visual? axes : axes-visual? function : (procedure-arity-includes/c 1) x : finite-real? id : symbol? baseline : finite-real? = 0 opacity : opacity? = 1 stroke : any/c = "gray" stroke-width : (and/c finite-real? (>=/c 0)) = 2
Creates the axes-aware vertical projection from (x, baseline) to
(x, function(x)). On a logarithmic y axis, the default baseline is the
minimum visible y value rather than zero.
procedure
(horizontal-line-to-graph axes function x #:id id [ #:baseline baseline #:opacity opacity #:stroke stroke #:stroke-width stroke-width]) → path-visual? axes : axes-visual? function : (procedure-arity-includes/c 1) x : finite-real? id : symbol? baseline : finite-real? = 0 opacity : opacity? = 1 stroke : any/c = "gray" stroke-width : (and/c finite-real? (>=/c 0)) = 2
Creates the horizontal projection from (baseline, function(x)) to the
graph point. On a logarithmic x axis, the default baseline is the minimum
visible x value.
procedure
(tangent-line axes function x #:id id [ #:dx dx #:length length #:opacity opacity #:stroke stroke #:stroke-width stroke-width]) → path-visual? axes : axes-visual? function : (procedure-arity-includes/c 1) x : finite-real? id : symbol? dx : (and/c finite-real? (>/c 0)) = 1/100 length : (and/c finite-real? (>/c 0)) = 2 opacity : opacity? = 1 stroke : any/c = "crimson" stroke-width : (and/c finite-real? (>=/c 0)) = 3
Estimates a tangent with the symmetric numeric difference at x. The
visible segment has world-space length and is centred on the graph
point. It is a numeric approximation, not symbolic differentiation.
procedure
(secant-line axes function x dx #:id id [ #:opacity opacity #:stroke stroke #:stroke-width stroke-width]) → path-visual? axes : axes-visual? function : (procedure-arity-includes/c 1) x : finite-real? dx : (and/c finite-real? (not/c zero?)) id : symbol? opacity : opacity? = 1 stroke : any/c = "darkorange" stroke-width : (and/c finite-real? (>=/c 0)) = 3
procedure
(secant-slope-group axes function x dx #:id id [ #:opacity opacity #:secant-stroke secant-stroke #:guide-stroke guide-stroke #:stroke-width stroke-width #:marker-radius marker-radius]) → group-visual? axes : axes-visual? function : (procedure-arity-includes/c 1) x : finite-real? dx : (and/c finite-real? (not/c zero?)) id : symbol? opacity : opacity? = 1 secant-stroke : any/c = "darkorange" guide-stroke : any/c = "gray" stroke-width : (and/c finite-real? (>=/c 0)) = 3 marker-radius : (and/c finite-real? (>/c 0)) = 1/10
Builds a secant, endpoint markers, dashed Δx/Δy legs, and
labels. Its stable children are named from id: id-secant,
id-delta-x, id-delta-y, id-first-point,
id-second-point, and the two corresponding label names.
procedure
(area-under-graph axes function #:id id [ #:x-min x-min #:x-max x-max #:baseline baseline #:sample-count sample-count #:opacity opacity #:fill fill #:stroke stroke #:stroke-width stroke-width]) → path-visual? axes : axes-visual? function : (procedure-arity-includes/c 1) id : symbol? x-min : (or/c finite-real? false/c) = #f x-max : (or/c finite-real? false/c) = #f baseline : finite-real? = 0 sample-count : (and/c exact-integer? (>=/c 2)) = 101 opacity : opacity? = 2/5 fill : any/c = "cornflowerblue" stroke : any/c = #f stroke-width : (and/c finite-real? (>=/c 0)) = 0
Samples one finite function and closes the result to baseline. The
result copies the current axes transform and uses one closed path subpath.
procedure
(area-between-curves axes first-function second-function #:id id [ #:x-min x-min #:x-max x-max #:sample-count sample-count #:opacity opacity #:fill fill #:stroke stroke #:stroke-width stroke-width]) → path-visual? axes : axes-visual? first-function : (procedure-arity-includes/c 1) second-function : (procedure-arity-includes/c 1) id : symbol? x-min : (or/c finite-real? false/c) = #f x-max : (or/c finite-real? false/c) = #f sample-count : (and/c exact-integer? (>=/c 2)) = 101 opacity : opacity? = 2/5 fill : any/c = "mediumpurple" stroke : any/c = #f stroke-width : (and/c finite-real? (>=/c 0)) = 0
Samples two finite functions over one domain and returns their closed filled
band in axes-local geometry.
procedure
(riemann-rectangles axes function #:id id [ #:x-min x-min #:x-max x-max #:count count #:baseline baseline #:opacity opacity #:fill fill #:stroke stroke #:stroke-width stroke-width]) → path-visual? axes : axes-visual? function : (procedure-arity-includes/c 1) id : symbol? x-min : (or/c finite-real? false/c) = #f x-max : (or/c finite-real? false/c) = #f count : exact-positive-integer? = 8 baseline : finite-real? = 0 opacity : opacity? = 2/5 fill : any/c = "seagreen" stroke : any/c = "darkgreen" stroke-width : (and/c finite-real? (>=/c 0)) = 1
Creates one closed midpoint rectangle per display-space interval. On a
logarithmic x axis the rectangles are evenly spaced in log display coordinates,
not by raw numeric width.
58.15.2 Deterministic Traced Paths
procedure
(traced-path phase position #:id id [ #:start-time start-time #:sample-count sample-count #:trail-length trail-length #:dissipate? dissipate? #:minimum-opacity minimum-opacity #:opacity opacity #:stroke stroke #:stroke-width stroke-width]) → derived-visual? phase : (or/c symbol? scene-parameter?) position : (-> derived-context? finite-real? vec2?) id : symbol? start-time : finite-real? = 0 sample-count : (and/c exact-integer? (>=/c 2)) = 121
trail-length : (or/c false/c (and/c finite-real? (>=/c 0))) = #f dissipate? : boolean? = #f minimum-opacity : opacity? = 0 opacity : opacity? = 1 stroke : any/c = "crimson" stroke-width : (and/c finite-real? (>=/c 0)) = 3
Creates a locus from an explicit scalar scene value. At every sampled frame,
Animate calls position at deterministically spaced times from
start-time to the current value of phase. Therefore a frame
at time t is independent of previously rendered frames; this is unlike a
mutable frame-history trail. The procedure receives the same read-only derived
context as other derived Visuals and must return a vec2 for every
sampled time.
With trail-length, the interval instead begins at the larger of start-time and current phase minus that length. With dissipate?, the resolved trace is an ordinary group of consecutive path segments whose opacity rises from minimum-opacity to opacity; otherwise it is one ordinary path Visual. No automatic tracking of arbitrary Visual motion or adaptive/discontinuity sampling is attempted.