Field / stage side Line of interest First row Row Center Radius Visual angle Isopter (constant angle) Straight row Sagitta Setback Row length Seat Worst seat Axis Plan view Fan method Isopter arc Straight row Within limits Check Not valid Row Radius Length Angle at center Angle at worst seat Seats Status rows seats deg The line of interest must be longer than zero. The first-row distance must be greater than zero. The available row length must be greater than zero. The seat pitch must be greater than zero. The fan multiplier m must be greater than zero. Row {n} would cross the line of interest: reduce d or increase m. Row {n} would sit behind the fan centre. Reduce the number of rows or increase m. The first row is closer than the {max}° limit. Minimum distance is {bmin}. The worst seat of row {n} sees only {val}; the target minimum is {min}. Head clearance fails at the worst seat of row {n}: sin φ = {val} and e/a = {ratio}. The isopter only spans {liso}; the row was trimmed from {d}. Increase b or shorten d. A straight row meets both the head-clearance and the angle criteria, so no arc is needed. The straight row leaves the worst seat below the angle target, so the isopter arc is used. Head clearance rules out a straight row, so the isopter arc is used. In a fan the angle always drops towards the ends. Compare the centre and worst-seat columns. All criteria are met for the current input. Enter valid values to see the rows. The arc end would be closer than {bc} to the line and was trimmed there. Horizontal Isoptic - Technical Report Method Units Date Input parameters Plan diagram Row-by-row results Notes Generated locally in the browser. No data is uploaded. Parameter Value Preset applied. DXF file generated. PDF report generated. Generating... Something went wrong while generating the file. The PDF library could not be loaded. Check your connection. Fix the input values before exporting.

Horizontal Isoptic

Plan-view seating geometry for auditoriums and stadiums. Choose a method, enter the parameters and the diagram, table and CAD file update as you type.

Horizontal only
Plan view
Metric / ft-in

The fan method

Every row is an arc of a circle drawn from one fan centre P placed m·c behind the line of interest. Because all rows share the same angular span β, the rows open out as they move back: this is the classic layout for drama theatres, lecture halls and small auditoriums where the stage opening c is narrow and the room widens quickly.

The angle is at its best at the centre and drops towards the ends, so both values are reported. The head-clearance check compares the angle between the sightline and the row with the ratio e/a.

Drama theater
Lecture hall
Community auditorium
Conference room

P = m · c R₁ = P + b

fan centre and first-row radius

β = d / (2 · R₁) Lₙ = 2 · Rₙ · β

half angle of the fan and length of row n

Rₙ = R₁ + (n − 1) · Δ N seats = ⌊Lₙ / a⌋

concentric rows and seat count

θ = 2 · atan[ (c/2) / (b + (n−1)·Δ) ]

visual angle at the centre of the row

sin φ ≥ e / a b ≥ b_min = (c/2) / tan(θmax/2)

head clearance and first-row limit

Plan view of the fan: centre P, radius R, half angle β, first rows at b and c.

Units

Presets

Geometry

cStage opening / line of interest

ft in m

bDistance to the first row

ft in m

dAvailable length of the first row

ft in m

Seating

aSeat pitch

ft in m

eHalf head width

ft in m

ΔRow depth

ft in m

NNumber of rows

Criteria

θ_maxMax. angle at the first row

θ_minMin. angle at the worst seat

mFan multiplier (P = m·c)

R—First-row radius

P—Centre behind the line

θ—Angle at the centre

θe—Angle at the worst seat

b_min—Minimum first-row distance

L_max—Straight-row limit

Fan method Within limits

Row-by-row results

Row	R	L	θ center	θ worst	Seats	Status

The drawing is a plan view; vertical sightlines, eye height and C values are outside the scope of this page. Angles are total horizontal angles subtended by the line of interest.

The isopter arc

The isopter is the locus of the points that see the whole line of interest under the same angle. For a straight line it is a circular arc through both ends of the line of interest, with radius R and centre P behind it. Every seat on that arc has the same visual angle, so the ends of the row are not penalised as they are in a fan.

The first row is placed at b. A straight row is used when the available length d fits inside the head-clearance limit L_max and the worst seat still reaches θmin; otherwise the isopter arc is drawn. Rear rows are concentric offsets of the same arc, which is how a stadium bowl grows.

Stadium
Sports arena
Amphitheater
Bowl seating

R = [ b² + (c/2)² ] / (2 · b)

isopter radius

P = R − b θ = 2 · atan[ (c/2) / b ]

centre behind the line and angle on the arc

L_max = c + 2·b·√(a² − e²) / e

longest straight row allowed by head clearance

L_iso = 2 · R · (π − θ)

arc length between both ends of the line

STRAIGHT if d ≤ L_max and θ_end ≥ θmin, else ARC

decision rule applied by the page

Plan view of the isopter: centre P, radius R, sagitta b and the constant angle θ.

Units

Presets

Geometry

cField width / line of interest

ft in m

bDistance to the first row

ft in m

dAvailable row length

ft in m

bcClearance line / setback (0 = none)

ft in m

Seating

aSeat pitch

ft in m

eHalf head width

ft in m

ΔRow depth

ft in m

NNumber of rows

Criteria

θ_maxMax. angle at the first row

θ_minMin. angle at the worst seat

R—Isopter radius

P—Centre behind the line

θ—Angle at the centre

θe—Angle at the worst seat

L_max—Straight-row limit

L_iso—Arc length between the ends

Isopter arc Within limits

Row-by-row results

Row	R	L	θ center	θ worst	Seats	Status

The drawing is a plan view; vertical sightlines, eye height and C values are outside the scope of this page. Angles are total horizontal angles subtended by the line of interest.

eXcco

Horizontal Isoptic

The fan method

Row-by-row results

The isopter arc

Row-by-row results

Fan method — parameters

Geometry

Seating

Criteria

Reading the drawing

Isopter arc — parameters

Geometry

Seating

Criteria

The decision