MacCready Speed-to-Fly: Glider & Paraglider Polar Calculator

The calculation is derived here from cycle time using the classical MacCready construction (MacCready, 1958; Cochrane's teaching examples). Sink $s$ is positive downward; the drawing plots $-s$.

Use true airspeed (TAS), with sink rates for the same density, weight and configuration. Convert all polar speeds to m/s. Fit $s(v)=av^2+bv+c$ exactly through the three supplied points using divided differences. Named steps show the slopes and coefficients. The fit is accepted only if its curvature and minimum sink within the fitted interval are positive. It must also satisfy $s(v)/v \le 0.20$ everywhere. For convex $s(v)$, $s(v)-kv$ is convex, so endpoint bounds imply the same bound throughout the interval. Thus $r_{max}=\max(s_1/v_1,s_3/v_3)$ gives the exact maximum sink/TAS; the two endpoints suffice for both physical and small-angle validation.

The exact horizontal airspeed is $u=\sqrt{v^2-s(v)^2}$. This classical construction approximates $u$ by TAS $v$. The chosen limit $s/v\le0.20$ corresponds to a glide angle at most 11.54° and a geometric overestimate $v/u-1$ at most 2.0621% at any fixed cruise speed. The cutoff is this tool's stated approximation limit, not an aircraft limit.

For a glide distance $D$, $t_g=D/v$, altitude lost is $sD/v$, and restoring that altitude at net climb $m$ takes $t_c=sD/(vm)$. Therefore

For an interior optimum, differentiating gives

Clamp this speed to the first and third polar speeds. A nonpositive radicand selects the lower endpoint. At an interior solution, the line from $(0,m)$ to $(v_*,-s(v_*))$ is tangent; its zero-height intercept is $V_{XC}$. At an endpoint it is a glide-and-climb secant. Best glide follows the same construction with $m=0$.

Example data are reference-density calculations at XCSoar density ratio 1, where IAS and TAS coincide in this idealization; they are not altitude-corrected. Data: attached XCSoar snapshot s3, p. 1 (ASW-27) and p. 4 (generic paraglider); signs are reversed from the source's negative sink convention. Cochrane, Safety Glides, attached s2 p. 1, supplies independent rounded ASW-27 speed and cycle-speed comparisons using a different, flight-manual polar.

• Straight, steady flight in still air between thermals. Polar speeds are true airspeeds. Horizontal speed is approximated by TAS only while sink/TAS is no greater than 0.20 throughout the fitted interval.
• The entered climb is the achieved net average climb of the aircraft. Do not subtract circling sink again.
• Glide and climb return to equal altitude; climb has no horizontal displacement. Thermal search and transitions have no additional time penalty.
• Custom points must pair TAS with sink rates measured or corrected to the same density, weight and configuration. Consistent indicated airspeeds at altitude are unsuitable; this tool does not convert IAS to TAS.
• Presets use the XCSoar reference polar at density ratio 1 (standard reference density), where IAS and TAS coincide in this model. Their results apply at that reference density and mass; no altitude, mass, ballast or bugs correction is applied.
• Example endpoints delimit available polar data, not aircraft operating limits. The generic paraglider profile is illustrative.
• Equations are derived here; the supplied data and external comparisons are separately sourced. Plot axes remain explicitly labelled in metric units; input and result units can be switched in the viewer.

Training only. Do not use these results for navigation, terrain clearance, final glides or in-flight speed commands. Respect the aircraft or wing flight manual and instructor guidance.

Only the interval between the first and third speeds is fitted; there is no extrapolation. A boundary warning means the unrestricted optimum is outside that interval. A line at the boundary need not be tangent. Convexity and positive-sink checks reject unsuitable custom fits. Sink reaching or exceeding TAS is physically invalid. The stricter small-angle limit rejects every fit with sink/TAS above 0.20 anywhere in the interval; both checks apply to the whole curve through its endpoint bounds. Correct the first or third speed/sink pair, checking units and matching density, when these errors appear. Passing the checks does not validate measured data or certify an aircraft.

At zero expected climb, average cycle speed is shown as its zero limit; a finite repeating altitude-neutral cycle is impossible. Best-glide speed is still meaningful for the stated idealization.

The displayed cycle speeds and glide ratios use TAS in place of horizontal speed. Their geometric overestimate at a fixed cruise speed is at most 2.0621% within the enforced range; the quadratic fit and operational assumptions introduce additional uncertainty. IAS data at altitude require an external density correction of both the speed and the polar sink rates before use.

Wind, lift and sink between thermals, gust response, safety margins, thermal availability and uncertainty are outside the model. Idaflieg notes that measurements made with different procedures are not directly interchangeable. Its reports have separate publication conditions; no report figure or table is reproduced here.

The ASW-27 comparisons allow 2–3% for different polar provenance and table rounding. This is a model comparison, not an exact reproduction of Cochrane's flight-manual fit.