What is the relationship between K value and Vertical curve radius? At profile labels, it gives me Radius = k * 100. Where did this equation come from and what is Vertical curve radius cuz it is parabola K-Value. This value represents the horizontal distance along which a 1% change in grade occurs on the vertical curve. It expresses the abruptness of the grade change in a single value. Speed tables or other design tools often provide a target minimum K value

L = Length of vertical curve, ft. A = Algebraic difference between grades, % K = Horizontal distance required to effect . a 1% change in gradient . V = Design speed, m ph. K VALUE FOR SAG VERTICAL CURVE (Comfort Criteria — 3R Project . Figure 55-4 The resulting sag vertical curves for the recommended stopping sight distances for each design speed are shown in Figure 2-7 with the solid lines representing the rounded K values. The minimum length of vertical curves (both crest and sag) are expressed as approximately three times the design speed in miles per hour (L min = 3 V) or 0.6 times. K-Value This value represents the horizontal distance along which a 1% change in grade occurs on the vertical curve. It expresses the abruptness of the grade change in a single value. Speed tables or other design tools often provide a target minimum K value

- section where this K-value is exceeded, carefully evaluate the drainage design near the apex. 6. Alignment Coordination. On rural facilities where crest vertical curves and horizontal curves occur at the same location, use the K-values in Figure 30-2A to ensure that the horizontal curve is visible as drivers approach the vertical curve
- K Value in a VERTICAL CURVE. FYI Everyone. Example. G1 = -1%. G2 = 3%. So the algebraic difference = 4%. The formula is also correct. Remember you NEVER can have a negative algebraic difference
- ology is introduced: g 1 is the slope (percent) of th
- desirable K value for a crest vertical curve is 405 for English units or 120 for metric units. Now solve for L, the desirable length of the curve: L = K × A = 405 × 5 = 2025 feet, or L = 120 × 5 = 600 meters. Sight Distance on Vertical Curves 6D-5 Design Manual Chapter
- within a profile context. In addition, sets the default Vertical Curve Parameter Format (Radius, K Value or M Value). Use the Profile settings to control the precision and display/input formats of slopes and ratios within a profile context. Click the Vertical Curve
- e the length in feet of the vertical curve that will provide
- imum length of vertical curve

As you are not using the OpenRoads vertical geometry tools, the Civil formatting setting does not apply here. The 'native' PowerCivil vertical geometry tools seem to allow only Length, r, or K values, so in your case the easiest thing to do is to divide your Radius by 100, and enter as the K value Paying more attention to the drainage design when **value** of **K** >167 The length of **vertical** **curve** can be computed by using **K** **values** in both crest and sag **vertical** **curves**. Minimum length of a crest **vertical** **curve** is equal 3 time the design speed (only for English Unit) 2 Lmin = A U K /100 a Where A = algebraic difference between grades in percent U = speed of the train km/h K = 0.077, conversion factor to give Lmin in m a = vertical acceleration, m/ s2 = 0.03 m/s2 for freight trains = 0.18 m/ s2 for passenger trains However, the length of any vertical curve cannot be less than 30 Designing a Curve to Pass Through a Fi d P i tFixed Point • Given: g 1,g 2, VPI station and elevation a point (P)VPI station and elevation, a point (P) elevation and station on the curve. • Required: You need five values to design a curve:Required: You need five values to design a curve: g 1,g 2, VPI station and elevation, and curve length. The only missing value is the length of the.

- The lengths of both crest and sag vertical curves are controlled by the available sight distance. K Values are calculated using the same equations as in Chapter 3, Section 4. Design Ks for both crest and sag vertical curves are shown on Table 8-9
- imum K value for small values of algebraic differences in grade (A) result in small curve lengths or no curve length at all because Chapter 2—Alignments Section 2B-1—Vertical Alignmen
- Enter the values x, g 1, g 2, and e pvc in the vertical curve elevation calculator to get the elevation of the point of vertical tangency. Use the Calculate button to find the final value in the vertical curve k value calculator
- To add a free parabolic vertical curve between entities. Add a free parabolic vertical curve between two entities by specifying a parameter. Select a pass-through point in the drawing or specify a K value, length, or radius to define the curve. The illustration shows a free parabolic curve between entities 1 and 2, passing through point 3
- 7 Nomenclature For Vertical Curves G1 & G2 Tangent Grade in percent A The absolute of the Algebraic difference in grades in percent BVC Beginning of Vertical Curve EVC End of Vertical Curve VPI Vertical Point of Intersection L Length of vertical curve D Horizontal distance to any point on the curve from BVC or EVC E Vertical distance from VPI to curve

Was the correct superelevation rate chosen for each curve? Identify curves with insufficient superelevation. Stopping Sight Distance: : GB Tables 3-1, 3-2, 3-34: Is stopping sight distance available along the horizontal alignment and for crest vertical curves? Use k values to assist calculation. Maximum Grade: : GB Tables 5-2, 6-2, 7- K-Value for Vertical Curves When creating a vertical curve for a profile of a road in LD 2006, I select FG Vertical Curve and then K-Value. Therefore, I select the grade in grade out (AD=12.75) and enter a k-value of 56. LD 2006 creates a vertical curve length of 178.56. It should be 714 • When a vertical curve takes place partly or wholly in a horizontal curve, coordinate the two as discussed in 1220.03. 1220.02(2) Minimum Length of Vertical Curves The minimum length of a vertical curve is controlled by design speed, stopping sight distance, and the change in grade

- New Shorter Vertical Curves • Old railway formula developed in 1880's for hook and pin couplers in those days • Present day couplers can accommodate shorter vertical curves • New formula developed in recent years: L = 2.15 V2 D / A V = train speed in mph D = algebraic difference of grade in decimal A = vertical acceleration in ft.
- ed graphically. 201.2.2 Vertical Stopping Sight Distance . The sight distance on crest vertical curves is based on a driver's ability to see a 2.0 ft. high object in the roadway without being blocked out by the pavement surface
- 3.5.7 Lowest and highest point of vertical curve. From the element of a vertical curve, we can write, Length of a curve (L) = R * N. Where, R is the radius of the curve and N is the deviation angle. Then, R = L / N. The highest point is calculated in a summit curve whereas the lowest is calculated in a valley curve

Table A 15: Minimum EDD crest vertical curve (K value) for sealed roads for the Norm-Day base case using an object height of 0.8 m (S < L) Table A 16: Minimum EDD crest vertical curve (K value) for sealed roads for the Norm-Day base case using an object height of 1.25 m (S < L K = a constant value for the design speed. The rate of change of grade along a vertical curve is constant, and is measured by dividing the algebraic difference between the grades by the length of the curve in feet [meters]. This value gives the percent change in grade per horizontal feet [meters] of curve According to the new method, the length of a vertical curve is calculated as follows: L = RQ (13.30) where L is the length of the vertical curve, R is the radius of the vertical curve as per the existing provisions given in Table 13.8, and Q is the difference in the percentage of gradients (expressed in radians) The K Value is a measure of the rate of change of curvature with distance (mathematically defined as the parabola length divided by the change in the gradient from the start to end point). A parabola does not have a true radius. The R Value is the K Value times 100 K Value. It is termed as a length of vertical curve divided by difference of algebraic in grade. It is nothing but a horizontal distance in feet (m) needed to make 1 % change in gradient. Intend controls for vertical curves shall be known in terms of vertical curvature K expressed as follows

Known values: Curvature radius H ~ R (positive for concave, negative for convex vertical curve), grade in S 1 and grade out S 2 (positive for ascending , negative for descending slopes), elevation of start point A of vertical curve H A Solution: Elevation of point K at a distance x from curve start point A y K, grade value at point K S K The vertical distance of point K from point A is calculated K values needed to produce vertical curves that will provide the required sight distances (i.e., B-1 curve sight distances) both for passenger cars and for trucks are generally greater than Green Book K values used for vertical curve design. Therefore, devel

Vertical curve elements (VPC = Vertical Point of Curvature; VPT = Vertical Point of Tangency). Factors to be considered in the selection of a vertical curve are: Stopping Sight distance S: On crest curves, S is a function of overall design speed of the road and driver's comfort. On most forest roads with design speeds from 15 km/hr to 30 km/hr. preserving developed properties and community values. 2. In alignment developed for a given design speed, the minimum radius of curvature for that speed should be avoided wherever practical. 3. Consistent alignment should always be sought. 4. For small deflection angles, curves should be sufficiently long enough to avoid the appearance of a. * The closer the individual curve's k values, the smoother the transition between them, particularly when there is a large grade change over a short distance*. b. Curve Equation. For an equal tangent vertical curve, we set up and solved Equations B-4 and B-5 through B-7

curve widening values on two-lane pavements (one- or two-way) for open highways. These values are a function of the degree of curve, the pavement width, and the design speed of the roadway. Table 3 presents the curve widening values for cases where the widening is 2.0 feet or greater. Table 3. Design Values for Pavement Widening on Highway. ** Draw a diagram with the values of your solution**. 5. A crest vertical curve on a 2-way, 2-lane road is designed for 100 km/hr. The grades are 1.6% and - 1.4%. If the tangents intersect at metric station 2 + 000.00 (where 2 represents 2 km and 000.00 is 0 meters) and at an elevation of 100 meters above sea level, determine the stopping sight. VERTICAL CURVE FORMULAS L = Length of Curve : g 2 = Grade of Forward Tangent PVC = Point of Vertical Curvature : a = Parabola Constant PVI = Point of Vertical Intersection : y = Tangent Offset PVT = Point of Vertical Tangency : E = Tangent Offset at PVI g 1 = Grade of Back Tangent.

- ed from max rate of superelevation and max side friction factor y Equation found on pg. 133 and pg. 143 y Can use this equation to solve for R_
- e the length in feet of the vertical curve that will provide
- Certain combinations of horizontal and vertical curves can result in an apparent distortion in the alignment or grade although the horizontal and vertical curves comply to the design standards outlined in Table A.7. Although this guide does not identify all concepts of good design form, Figures B-3.2a through B-3.2
- Minimum Length of Vertical Curves (8.6.7) Minimum curve lengths (Table 8.10) increased by 50% when approach is straight and vertical curve is within driver's view for at least 500 m. On resheet projects it is acceptable to adopt the shorter vertical curves in Table 8.11. Vertical alignment 5

- e the length of vertical curve
- g a smaller object height for the preferred values in Table 5C-1.01 are higher required K values and longer vertical curves. 2. Sight Distance on Horizontal Curves: The horizontal alignment must provide at least th
- e the design speed of the curve
- An example for finding elevation on vertical curves.Follow Simmy Sigma on Twitter: @simmysigmaMake sure to like the Official Simmy Sigma Facebook Page here..

44-3C(1) Stopping Sight Distance Check Using K-Values, Sag Vertical Curve 44-3D K-Values for Sag Vertical Curves (Decision Sight Distance - Passenger Cars) 44-3D(1) Sight Distance at Undercrossings 44-3E Vertical Curve Definitions 44-3F Symmetrical Vertical Curve Equations 44-3G Vertical Curve Computations (Example 44-3.1) 44-3H Unsymmetrical. Sirong Yi, in Dynamic Analysis of High-Speed Railway Alignment, 2018. Abstract. The minimum curve radius of a single high-speed railway line should ensure that when the superelevation is set to ultrahigh maximum, the unbalanced centrifugal acceleration does not exceed the kinetic characteristics of the allowable value. For high-speed railway wherein the high-speed train and the medium-speed.

- ation of Minimum Radius Vertical Curves 12-11 12.4.1 Minimum Radius Crest Vertical Curves 12-11 Appearance Criterion 12-11 Sight Distance Criteria 12-11 12.4.2 Minimum Radius Sag Curves 12-11 Appearance Criterion 12-1
- e crest vertical curve length. See charts for 20 mph (PDF), and 30 mph (PDF) design speed crest vertical curve lengths. Horizontal Curves. Radius based on Lean Angle (MnDOT BFDM) 74 ft
- Radius of Horizontal Curve w/ Sight Obstruction Hydrology. Design Control for Crest Vertical Curve When S is Less Than L. U.S. Customary English Units. See Roadway Design Manual Figure 4-I. This program calculates sight distance on a crest vertical curve when the sight distance is less than the length of curve. Using the calculated.
- Symmetric parabolic vertical curves#VerticalCurves #HighwayDesign #DanielFindle
- Note: S and K values given above and used in calulations are the rounded design values as given in Exhibit 3-76, pg 274 of AASHTO's A Policy on Geometric Design of Highways and Streets, 2001. Crest Vertical Curves October 22, 2004 Minimum Length of Vertical Curves of Stopping Sight Distanc

* How do I set the K Value in SS3? Answer: First, you will need to set the Vertical Curve Parameter to K Value: In the MicroStation menu, select Settings > Design File > Civil Formatting*. In Profile Settings, set the Vertical Curve Parameter to Kvalue. Now, when you create your vertical curve, enter the K Value as the Vertical Curve Parameter Vertical Curve Length - A Transition Curve. Vertical curves are important elements in geometric design of highways, providing transition between two sloped roadways. These are parabolic curves and are assigned stationing based on horizontal axis. Feedback The geometric design of roads is the branch of highway engineering concerned with the positioning of the physical elements of the roadway according to standards and constraints. The basic objectives in geometric design are to optimize efficiency and safety while minimizing cost and environmental damage. Geometric design also affects an emerging fifth objective called livability, which is. This is the exact expression for superelevation. But values of e and f are very small so it can be simplified as, Therefore, v 2 / (g*R) = (e + f) Where, e is the rate of superelevation and f is the coefficient of lateral friction, v is the velocity of the vehicle in m/s, R is the radius of the curve and g is the acceleration due to gravity in.

Maximum value of gradient compensation = %, R= radius of horizontal curve. Vertical Curves Generally two types of vertical curves are there to adopt for the vertical alignment. They are Summit curve; Valley curve; Summit curve Summit curve is a vertical curve adopted mainly when the gradient is upwards. In case of summit curve simple parabola. designing for passing sight distance. If this crest vertical curve was connected by lengthy tangents sections extending at +1.0% and -1.0%, instead of short tangent sections connecting to sag vertical curves as shown above, then this crest vertical curve would not meet the minimum passing sight distance for 60 mph design speed The values of K for sag vertical curves range from 10 ft per percent difference in grade at 15 mph to 231 ft per percent difference in grade at 80 mph. 2.1.3 Horizontal Sight Distance Obstructions Horizontal sight distance obstructions are opaque objects that limit the driverâ s view of the roadway ahead and are located on the inside of. A = absolute value of the differences in grades () expressed as a percentage. The general form of the parabolic equation, as applied to vertical curves, is. Where, y = roadway elevation at distance x from the beginning of the vertical curve in ft (m) x = distance from the beginning of the vertical curve in stations or ft (m

High Speed 1 (section 2) in the UK has a minimum vertical curve radius of 10,000 m (32,808 ft) and High Speed 2, with the higher speed of 400 km/h (250 mph), stipulates much larger 56,000 m (183,727 ft) radii. In both these cases the experienced change in weight is less than 7%. Rail well cars also risk low clearance at the tops of tight crests By definition, **K** (rate of **vertical** curvature) is the horizontal distance in ft (meters) required to affect a 1% change in the slope of a **vertical** **curve**. It has several uses including simplifying the computation of the high/low points of **vertical** **curves**. **K** is calculated as follows grades are the slope of the tangent line to the vertical curve. The values of grades are generally signified by g 1 and g 2. The point of vertical curvature (PVC) is associated to g 1 while the point of vertical tangency (PVT) is connected to g 2. The curvature grade is positive if it is upward and Radius of Second Curve R The following design aids and information may be used to develop a roadway design and prepare PS&E packages. Division tools are also available at left.Roadway GeometryThe following spreadsheets are intended to assist Highway Design professionals in completing lines and grades. All results should be verified by a Professional Engineer. Unless otherwise stated the spreadsheet Minimum curve length for appearance criteria e. Design K factor value and minimum vertical curve length from table 6. A sag vertical curve goes under a railroad underpass with a 14' clearance. The curve has an entering grade of -5% and a departure grade of 2%

2.1 Minimum Curve Radius based on the limit Value of Superelevation Minimum curve radius can be derived to take into account the limit value of superelevation. Equation of minimum horizontal curve radius depend on the superelevation value is given by American Association of State Highway and Transportation Officials' A Policy on Geometri * K Values for Adequate SSD US Customary Metric Rate of vertical curvature, Ka Rate of vertical curvature, KaDesign speed (mi/h) Stopping sight distance (ft) Calculated Design Design speed (km/h) Stopping sight distance (m) Calculated Design 15 80 9*.4 10 20 20 2.1 3 20 115 16.5 17 30 35 5.1 6 25 155 25.5 26 40 50 8.5 9 30 200 36.4 37 50 65 12.2. Chapter 3 Section 3-03 Vertical Alignment 3 2019 May. Vertical Curve Geometry: The grade line G is always expressed in terms of a percentage. When the grade rises in the direction of increasing stationing, G i

Therefore, designers use a series of curves to develop road alignments. Most highways are comprised of circular curves. A curve with a smaller radius will result in a sharper curve. Therefore, roads with higher speed limits need to be designed as a series of large-radius curves. The table below lists typical curve radii by road type 4.1.1 Recognize a function of two variables and identify its domain and range. 4.1.2 Sketch a graph of a function of two variables. 4.1.3 Sketch several traces or level curves of a function of two variables. 4.1.4 Recognize a function of three or more variables and identify its level surfaces. Our first step is to explain what a function of. of vertical curvature is equal to the length of the vertical curve (L) divided by the algebraic difference in the tangent grades (A) in percent (K=L/A). For a given design speed the minimum length of the sag vertical curve can be verified by determining the rate of vertical curvature and checking this value agains needed for a vertical curve to produce a -percent 1 change in gradient. Minimum K-values relate lengths of vertical curves to required sight distance on crest curves and to headlight beam distance on sag vertical curves. The design team should consider various options to satisfy criteriasight distance on vertical curves

Design Control for Crest Vertical Curves Design Rate of Vertical Curvature, (K Value) Design Control for Sag Vertical Curves Note: If the K Value is below that required for Design Stopping Sight Distance, the warning K Value does not meet AASHTO Design Standards will appear beside the K Value. K Value of Prop. Curve: 15.00 80.00 3.00 15.00 80. **K-Value**. This **value** represents the horizontal distance along which a 1% change in grade occurs on the **vertical** **curve**. It expresses the abruptness of the grade change in a single **value**. Speed tables or other design tools often provide a target minimum **K** **value** Also, vertical curves are sometimes described by K, the reciprocal of r. K is the distance in meters required to achieve a 1 percent change in grade. Verti-cal curves are classiﬁed as sags where g 2 g 1 and crests otherwise. Note that r (and hence the term rx2 2) will be positive for sags and negative for crests Vertical Curves. Vertical Curves are nice! I think they are easier to solve than Horizontal Curves because they are essentially parabola equations (which you should be very used to). This is the formula for vertical curves. is the elevation y at x horizontal distance into the curve from the start position. is the starting elevation at x = 0

- Vertical curve design in AASHTO Green Book will not apply for such a low speed driveway. Your main concern should be to design a vertical curve sufficient enough to safely pass larger vehicles (i.e. moving vans, pickups with trailers, etc.) through the vertical deflection without dragging or getting stuck. I would start with a K-value of '5'
- 3. • Avoid placing a low point of a sag vertical curve in cut sections. To drain these low points, additional construction cost may be incurred. 4. • Short humps in vertical alignment, that tend to drop out of sight, shall be avoided whether on a horizontal curve or a tangent alignment. 5
- If not, either the radius of the horizontal curve has to be increased or the speed has to be restricted to the safe value given in equation 1.6 which will be less than the design speed. =( +) (1.6) Appropriate warning sign and speed limit regulation sign are installed to restrict and regulate the speed to at such curves

- * The curvature of vertical curve should be as large as possible to provide for comfort and sufficient sight distance for safe stopping at design speed. The curvature should be derived from the appropriate K value where K = L (curve length in metres)/A (algebric difference in gradients (%)). Design speed Minimum K value for vertical cres
- imum grade requirement shall override the slope within the vertical.
- = Radius of the adjoining circular curve. D c = Degree of curve of the adjoining circular curve, based on a 100 foot arc (English units only). D = Degree of curve of the spiral at any point, based on a 100 foot arc (English units only). l = Spiral arc from the TS to any point on the spiral (l = Ls at the SC)

- The surveyor can establish curves of short radius, usually less than one tape length, by value to the I angle. ordinate bisects the central angle. 3-3 . FM 5-233 Degree of Curve (D
- By definition, K (rate of vertical curvature) is the horizontal distance in ft (meters) required to affect a 1% change in the slope of a vertical curve. It has several uses including simplifying the computation of the high/low points of vertical curves. K is calculated as follows
- e

Vertical Curve Calculation. The vertical point of curvature (PVC) and the vertical point of tangency (PVT) are located a horizontal distance of L/2 from the PVI. The PVC is generally designated as the origin for the curve and is located on the approaching roadway segment. The PVT serves as the end of the vertical curve and is located at the. Given the following K values, what is would be the base curve if fitted on K (all the values are in millimeters.) What would be the intermediate curve of a lens of base curve 8.9mm. (all the values are in diopters.) You get the idea. Conventional Rules of Thumb for Radius/Diopter Conversion Most Civil 3D vertical curve label styles contain several components to describe a vertical curve. Typically, these styles include: curve length, PVI station/elevation, curve start and end station/elevation, high (or low) point station/elevation, K value, etc., making them perfect when applied to most vertical curves. If a vertical curve label is split between profile views however, you may.

The speed/radius limitations are shown in the TDH page B4.1.It is possible to connect gradients without using vertical curves if they are of a similar gradient with less than 0.02% difference. These criteria are again shown in the TDH. iii. The design values regarding vertical curves should be used in the . following order: Normal values There are eight (8) vertical curves within the Study Area. Four (4) are crest curves and four (4) are sag curves. As long as the K-Values are met, or exceeded, the length of the curve can be less than the stopping sight distance. All curves within the Study Area exceed the required design K-values ** Draw also the approximate K(h)-curve for soil B**. K sat h K A and B A B θ θ sat θ θ sat air entry value The air entry value of a specific soil is determined by the radius of the largest pore. If this largest pore is relatively small, the air entry value will be relatively large. K sat B K A h air entry value

spiral segment to provide an a value equal to or less than the standard spiral for the sharper curve seeAppendix J . The a value is a measure of the rate of change of the curvature. (Change in Degree of curve x 100 / length of spiral). Broken back curves are curves in the same direction connected with a short segment of tangent R Radius OA = OB = OC L Length of Curve L = 0.0174533 R ∆ T Tangent Distance T = AV = R tan Curves are usually fitted to tangents by choosing a D (degree of curve) that will place the centerline of the curve on or slightly on or above the gradeline. Sometimes D is chosen to satisfy a limited tangent distance or a desired curve Vertical curves are used to provide gradual change between two adjacent vertical grade lines. The curve used to connect the two adjacent grades is parabola. Parabola offers smooth transition because its second derivative is constant. For a downward parabola with vertex at the origin, the standard equation is. x 2 = − 4 a y or y = − x 2 4 a

The maximum value of the above two is taken as grade compensation. But compensated grade can not be less than 4%. Where, R = Radius of curve in meter. Vertical Curve. Due to changes in grade in the vertical alignment of the highway, it is necessary to introduce a vertical curve at the intersections of different grades to smoothen out the. Approximate values for the total amount of curve widening required by a single unit design truck on a circular arc section of a two-lane two-way road are shown in Table 4.1. The amount of widening require d, on a per lane basis, can be calculated using the method detailed in Figure 4.5. Curve Radius (m) Total amount of traffic lane widening. ** This angle is called the degree of the curve (D)**. The relation between the radius and the degree of the curve may be determined as follows: Refer to fig 11.6: Let R= The radius of the curves in meters . D= The degree of the curve . MN= The chord, 30m long . P= The mid-point of the chord . The approximate relation holds good up to 5° curves §2.6.4 Horizontal Curve Radius : Text revised to clarify how . horizontal curve radii are measured. Exhibits 2-3, 2-3a, 2-4, Shoulder Width Vertical Clearance Horizontal Curve Radius Design Loading Structural Capacity Design criteria values should be established taking into consideration the Department's Contex

• (POC) POINT OF CURVE. The point of curve is any point along the curve. • (L) LENGTH OF CURVE. The length of curve is the distance from the PC to the PT, measured along the curve. • (T) TANGENT DISTANCE. The tangent distance is the distance along the tangents from the PI to the PC or the PT. These distances are equal on a simple curve Sag Vertical Curves. Sag vertical curves are curves that connect descending grades, forming a bowl or a sag. Designing them is is very similar to the design of crest vertical curves. Once again, the sight distance is the parameter that is normally employed to find the length of the curve The center is at the origin. The equation of a circle in standard form is: (x−h)²+ (y−k)²=r². Where r is the radius and (h,k) is center. If either -h or -k is missing, then its value must be 0. Thus, if both are missing the circle must be centered at the origin, (0,0). Comment on Just Keith's post The center is at the origin

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