Точное руководство по расчетам для изогнутого светодиодного дисплея: Длина дуги, Длина аккорда, и стрелка (От теории до практики)

In curved Светодиодный дисплей projects, Точный расчет длины дуги, Длина аккорда, and sagitta directly determines the display’s splicing precision, installation compatibility, and visual effect. Whether it’s an indoor curved conference screen, an outdoor curved advertising screen, or an immersive circular screen for a venue, a calculation deviation exceeding 10cm can lead to modules that won’t fit together, large gaps between frames, or even the need to re-customize the screen, resulting in wasted cost and project delays. This article will systematically explain the calculation logic for key dimensions of a curved LED display from four perspectives: measurement methods, formula principles, practical examples, and error control, to help engineers avoid common problems and achieve precise installation.

I. Core Concepts and Measurement Prerequisites: First, Understand the “Three Key Dimensions”

Before calculating, you must clearly define the three core dimensions of a curved LED display to avoid measurement errors caused by conceptual confusion.

1. Definitions of the Three Key Dimensions

• Длина аккорда (2a): The straight-line distance between the two end points of the arc. It’s a direct representation of the “horizontal span” of the curve. For example, the straight-line distance between the two anchor points of an outdoor curved advertising screen is the chord length. It’s usually represented as “2a” (where ‘a’ is half the chord length) to facilitate subsequent right-triangle calculations.
• Sagitta (b): The vertical distance from the midpoint of the chord to the highest point of the arc. It determines the “curvature” of the arc. The larger the sagitta, the steeper the curve; the smaller the sagitta, the flatter the curve (closer to a straight line). For example, the distance an indoor curved conference screen bulges out from its ends at the center is the sagitta.
• Длина дуги (L): The actual length of the arc itself, which represents the “total length” of the assembled LED modules. Since the arc is a curve, it cannot be measured directly with a tape measure. It must be calculated using the chord length and sagitta. It is also the core basis for determining the number of modules when customizing the screen (Number of modules = Arc Length ÷ Length of a single module).

2. 3 Key Preparations Before Measurement

• Tool Selection: Prioritize using a laser distance meter (accuracy ±1mm) to measure the chord length to avoid errors caused by tape measure stretching. For sagitta measurement, a “spirit level + tape measure” combination is required to ensure the measurement is perpendicular to the chord length (if tilted, the measured sagitta will be larger).
• Measurement Environment Requirements: Outdoor measurements should be done in calm weather to prevent the tape measure from swaying. In hot environments, wait for the laser distance meter’s temperature to stabilize before use (for every 10°C change in temperature, the accuracy may deviate by 0.1%). Indoor measurements require a flat surface to avoid chord length deviations caused by ground slopes.
• Measurement Point Marking: Clearly mark the two end points and the midpoint of the arc (e.g., with tape or a cross). It’s recommended to measure each point 3 times and take the average as the final data (e.g., if the chord length is measured three times as 200.2cm, 199.8cm, and 200.0cm, take the average of 200.0cm, so 2a=200cm).

II. Precise Measurement Methods for Chord Length and Sagitta: Avoid 3 Common Pitfalls

Chord length and sagitta are the fundamental data for calculating the arc length and radius. Measurement errors will directly magnify subsequent calculation deviations, so you must strictly follow the standard procedures and avoid the following pitfalls.

1. Chord Length Measurement: Eliminate the “Two-Points-Not-in-a-Straight-Line” Error

• Correct Measurement Steps:
  1. Determine End Points: Find the two outermost end points of the arc (Point A and Point B) and ensure the line connecting them covers the maximum horizontal span of the arc (if the end points are off, the chord length will be smaller).
  2. Horizontal Calibration: Use a spirit level to confirm that Point A and Point B are on the same horizontal plane (if the outdoor ground is uneven, place pads of the same height at the two points and then measure the distance between the tops of the pads).
  3. Laser Distance Measurement: Place the laser distance meter flush against the mark at Point A, aim it at the mark at Point B, keep the meter parallel to the line connecting the two points, and read the data. Repeat 3 times and take the average.
• Common Pitfalls and Avoidance:
  • Pitfall 1: Using a tape measure directly along the ground, ignoring the ground’s slope. For example, with a 3° slope and a 200cm chord length, the measurement can be off by about 10.4cm (sin3° × 200 ≈ 10.4).
  • Avoidance: Use a spirit level for calibration to ensure the measurement direction is horizontal, or use a laser distance meter with slope compensation.
  • Pitfall 2: Vague end point markings leading to different measurement points each time.
  • Avoidance: Use an indelible marker to draw a 5cm diameter cross at the end points, and use the intersection of the cross as the reference point for measurement.

2. Sagitta Measurement: Ensure “Perpendicular Midpoint” Accuracy

• Correct Measurement Steps:
  1. Find Chord Midpoint: Use a tape measure to find the midpoint (Point O) of the chord length (from A to B) (if the chord length is 200cm, Point O is 100cm from both A and B). Mark Point O.
  2. Perpendicular Positioning: Level the vertical bubble of a spirit level and place a tape measure along the vertical direction (perpendicular to the chord length). Align one end of the tape measure with Point O and extend the other end to the highest point of the arc (Point C), reading the distance OC, which is the sagitta ‘b’.
  3. Multiple Verifications: Measure the vertical distance at 10cm to the left and right of Point O (Points O1, O2). If the difference between O1C, O2C, and OC is ≤2mm, it indicates that Point C is the true midpoint of the arc. If the difference is too large, re-confirm the highest point of the arc.
• Common Pitfalls and Avoidance:
  • Pitfall 1: Measuring the sagitta in a direction that is not perpendicular to the chord length, resulting in a larger measurement. For example, a 5° tilted measurement of a 60cm sagitta will result in an actual sagitta of only 59.7cm (cos5° × 60 ≈ 59.7).
  • Avoidance: Use a right-angle ruler or spirit level to ensure the tape measure is perpendicular to the chord length. If necessary, use a laser line generator to project a vertical baseline.
  • Pitfall 2: Mistaking an “edge point of the arc” for the “highest point.” For example, if a module protrudes at the edge of the curved screen, you might mistakenly take this as Point C, leading to a smaller sagitta measurement.
  • Avoidance: Observe the overall curvature of the arc and feel it to confirm the highest point, or use a laser line generator to project a line along the chord length and find the point furthest from it.

III. From Chord Length and Sagitta to Arc Length and Radius: Formula Principles and Practical Examples

The core of curved LED display dimension calculation is based on the “right-angle triangle + arc relationship of a circle.” By using the chord length (2a) and sagitta (b) to construct a right-angle triangle, and then combining it with trigonometric functions and the circumference formula, you can deduce the radius (R) and arc length (L). The following steps break down the calculation process using a practical example (2a = 200cm, b = 60cm).

1. Step 1: Construct a Right-Angle Triangle to Calculate the Hypotenuse (c) Half of the chord length (a), the sagitta (b), and the “portion of the radius of the circle that the arc is a part of, minus the sagitta (R-b),” together form a right-angle triangle (as in Figure 1):

• Leg 1: a = Chord Length ÷ 2 = 200cm ÷ 2 = 100cm
• Leg 2: R – b (R is the radius of the circle the arc is a part of, which is currently unknown)
• Hypotenuse: c = R (because the hypotenuse is the line connecting the center of the circle to the end point of the arc, which is the radius) According to the Pythagorean theorem (the sum of the squares of the two legs of a right-angle triangle is equal to the square of the hypotenuse): a² + (R – b)² = R²
• Substituting the example values (a=100cm, b=60cm): 100² + (R – 60)² = R² Expanding the calculation: 10000 + R² – 120R + 3600 = R² Eliminating R² and rearranging: 13600 – 120R = 0 Solving for R: R = 13600 ÷ 120 ≈ 113.3cm (The radius of the circle the arc is a part of is approximately 113.3cm)
• Note: You must avoid the mistake of “directly calculating the hypotenuse using a and b and treating it as the radius.” The correct logic is to set up an equation using the Pythagorean theorem and solve for the radius R, not to directly calculate the hypotenuse of the right-angle triangle.

2. Step 2: Calculate the Central Angle of the Arc (θ) The central angle is the angle formed by the lines connecting the arc’s two end points to the center of the circle (as shown by ∠AOB in Figure 1). It determines the proportion of the whole circle that the arc occupies and is key to calculating the arc length.

• Calculation Logic: In the right-angle triangle (OAC), cos∠AOC = adjacent side ÷ hypotenuse = (R – b) ÷ R. ∠AOC is half of the central angle ∠AOB (because OC perpendicularly bisects AB, dividing ∠AOB into two equal angles), so: ∠AOB = 2 × ∠AOC
• Substituting the example values (R≈113.3cm, b=60cm): R – b = 113.3 – 60 = 53.3cm cos∠AOC = 53.3 ÷ 113.3 ≈ 0.47 Using a calculator’s “inverse cosine function (arccos)” to calculate ∠AOC: ∠AOC ≈ arccos(0.47) ≈ 61.9° Therefore, the central angle ∠AOB = 2 × 61.9° ≈ 123.8° (approximately 124°, consistent with the original case).
• Additional Note: If you use the sine function, sin∠AOC = a ÷ R = 100 ÷ 113.3 ≈ 0.88, which gives ∠AOC ≈ 61.6°. The results are similar (the difference comes from rounding decimals). Both methods work, but it’s recommended to base the calculations on the R value solved using the Pythagorean theorem to avoid cumulative errors.

3. Step 3: Calculate the Arc Length (L) Based on the Central Angle and Circumference The arc length is the actual length of the curve. It equals the “circumference of the entire circle multiplied by the ratio of the central angle to 360°.” The formula is: L = (θ ÷ 360°) × 2πR Where π is approximately 3.14, θ is the central angle (in degrees), and R is the radius.

• Substituting the example values (θ≈124°, R≈113.3cm):
  1. First, calculate the circumference of the circle: 2πR = 2 × 3.14 × 113.3 ≈ 711.5cm
  2. Calculate the arc’s proportion of the circle: 124° ÷ 360° ≈ 0.344
  3. Arc Length L = 0.344 × 711.5 ≈ 244.8cm (approximately 245.5cm, consistent with the original case, with the difference due to rounding the central angle decimal).
• Verification Logic: If you calculate it directly using “arc length per degree × central angle” (Arc length per degree = 2πR ÷ 360 ≈ 711.5 ÷ 360 ≈ 1.98cm/degree), the arc length L = 1.98 × 124 ≈ 245.5cm. The results are the same, so you can cross-verify using both methods to ensure accuracy.

4. Expanding Calculation Cases for Different Scenarios To cover more project needs, here are two additional common scenarios with calculation examples to help engineers apply the formulas flexibly.

• Scenario 1: Small-Curvature Curved Screen (Chord Length 300cm, Sagitta 30cm)
  • Step 1: a=150cm, b=30cm. Substitute into the Pythagorean theorem: 150² + (R-30)² = R² → 22500 + R² – 60R + 900 = R² → R = 23400 ÷ 60 = 390cm
  • Step 2: Central angle θ = 2 × arccos((390-30)/390) = 2 × arccos(360/390) = 2 × arccos(0.923) ≈ 2 × 22.6° ≈ 45.2°
  • Step 3: Arc length L = (45.2 ÷ 360) × 2 × 3.14 × 390 ≈ 0.125 × 2450.4 ≈ 306.3cm
• Scenario 2: Large-Curvature Circular Screen (Chord Length 500cm, Sagitta 200cm)
  • Step 1: a=250cm, b=200cm. Substitute into the Pythagorean theorem: 250² + (R-200)² = R² → 62500 + R² – 400R + 40000 = R² → R = 102500 ÷ 400 = 256.25cm
  • Step 2: Central angle θ = 2 × arccos((256.25-200)/256.25) = 2 × arccos(56.25/256.25) = 2 × arccos(0.22) ≈ 2 × 77.3° ≈ 154.6°
  • Step 3: Arc length L = (154.6 ÷ 360) × 2 × 3.14 × 256.25 ≈ 0.429 × 1610 ≈ 690.7cm

IV. Error Control and Practical Advice: Ensuring “Usable, Accurate” Results

Even with correct formula application, ignoring error control can still lead to installation problems. Here are some practical tips from three perspectives: “measurement error, formula application, and on-site verification.”

1. Measurement Error Control: Keep Deviation Within 5cm

• Chord Length Measurement: Use a laser distance meter with an accuracy of ≥±1mm. The average of 3 measurements should have a deviation of ≤3mm from the single largest value. If the chord length is over 500cm, it’s recommended to measure in segments (e.g., 2 segments, 3 times each) to avoid cumulative errors.
• Sagitta Measurement: Keep the vertical deviation ≤2°. The deviation of the measured value from the actual value should be ≤5mm. If the curved screen has module joints, you must take supplementary measurements at the joints to ensure the sagitta is the “true height of the screen surface” (not the frame height).
• Handling Extreme Scenarios: For extra-large outdoor curved screens (Длина аккорда > 10м), it’s recommended to use a total station for measurement, which can achieve an accuracy of ±2mm. Also, set up temporary reference points at the two ends and the midpoint of the arc to reduce environmental interference.

2. Formula Application Errors: Avoid 3 Calculation Traps

• Trap 1: Confusing the relationship between “sagitta” и “radius,” and directly using the hypotenuse of the right-angle triangle as the radius (e.g., mistakenly using c=116.62cm as R in the original case), leading to a radius calculation deviation (the correct R=113.3cm, a deviation of 3.3cm).
• Avoidance: Strictly set up an equation using the Pythagorean theorem to solve for R, instead of directly calculating the hypotenuse of the right-angle triangle.
• Trap 2: Not dividing by 2 when calculating the central angle, which doubles the central angle (e.g., using ∠AOC as ∠AOB), resulting in a doubled arc length calculation.
• Avoidance: When calculating the central angle, clearly understand that the “angle in the right-angle triangle is half of the central angle.” If necessary, draw a diagram to label the angle relationships.
• Trap 3: Using different numbers of decimal places for π (e.g., 3.1416) without a consistent approach, leading to inconsistent results.
• Avoidance: Use a consistent value for π (e.g., 3.14 or 3.1416), and keep 3 decimal places throughout the calculation. The final result should be rounded to 1 decimal place (e.g., 245.5cm) to facilitate module customization.

3. On-Site Verification: А “Last Line of Defense” Before Installation

• 1:1 Layout: For projects with complex curves (e.g., double curves, irregular shapes), use a chalk line to lay out the calculated dimensions 1:1 on the ground. Simulate the arc’s trajectory to check if it fits the on-site walls and frames.
• Module Pre-Splicing: Take 3-5 LED modules and splice them along the layout trajectory according to the calculated arc length. Check the gaps between the modules (they should be ≤1mm). If the gaps are too large, re-verify the chord length and sagitta measurement data.
• Radius Re-verification: Use a tape measure to measure the distance from the arc’s two end points to the center of the circle (if the center can be determined). If the distance deviates from the calculated R by ≤2cm, it indicates that the calculation is correct. If the deviation is too large, you must re-measure the chord length and sagitta.

V.. Summary: А “Core Process” for Curved LED Display Dimension Calculation

For easy application by engineers, the entire calculation process is summarized into a “4-step core method,” with a clear objective and verification logic for each step:

1. Measure Base Data: Use a laser distance meter + spirit level to measure the chord length (2a) and sagitta (b). Take the average of 3 measurements, with a deviation of ≤3mm.
2. Find the Radius (R): Set up an equation using the Pythagorean theorem, a²+(R-b)²=R², and solve for R, ensuring no sign errors in the calculation.
3. Calculate the Central Angle (θ): Use the inverse cosine function in the right-angle triangle to calculate half of the central angle, then double it to get θ. Round the result to 1 decimal place.
4. Get the Arc Length (L): Use the formula L=(θ÷360°)×2πR to calculate the arc length, and cross-verify with the “arc length per degree × θ” method. A deviation of ≤1cm is acceptable.

By following this process, you can ensure that the arc length, Длина аккорда, and sagitta calculations for a curved LED display are accurate, providing reliable data for module customization, frame design, and on-site installation, thereby avoiding project risks caused by dimension deviations. In actual projects, if you encounter an irregular curve (e.g., an elliptical curve), you will need to introduce elliptical equations. It’s recommended to collaborate with a structural engineer and the screen manufacturer to develop a solution that ensures the final result meets the design expectations.