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# TI-84 Plus CE and Casio fx-5800P: Highway Transition Spiral

TI-84 Plus CE and Casio fx-5800P:  Highway Transition Spiral

Introduction

For any given parameters of a highway transition spiral:

PI:  point of intersection of the vertices (X coordinate, Y coordinate; Program Variables A, B)
∆:  intersection angle  (Program variable I)
D_C:  degree of the curve (Program variable D)
L_S:  length of the spiral (Program variable L)

Outputs:

Z:  Angle between the radii of spiral at TS and SC
TS:  point of intersection of main tangent and approach spiral   (Program Variables U, V)
SC:  point of intersection of main tangent and circular curve
ST:  short tangent

Formulas:

Z = L * D / 200

Z is in degrees
Convert Z to radians:  Q = Z * π /180

x = L * (1 – Q^2/10)
y = L * (Q/3 – Q^3/43)
k = x – R * sin Z
p = y – R * (1 – cos Z)
T = (R + P) tan (∆/2) + K
S = Y * csc Z = Y / sin Z

Coordinates of TS:
[ PI_X – intg(T/100),  PI_Y – 100 * frac(T/100)]

Coordinates of SC:
[ TS_X + intg(L/100), TS_Y + 100 * frac(L/100)]

TI-84 Plus CE Program HYSPIRAL
(program to be typed)

Degree
“EWS 2019-08-03”
Disp “HWY SPIRAL”
Input “PI X: “,A
Input “+ PI Y: “,B
Input “INT-ANGLE: “,I
Input “LENGTH: “,L
Input “DEGREE: “,D
L*D/200→Z
100/(D*π/180)→R
Zπ/180→Q
L(1-Q²/10)→X
L(Q/3-Q³/43)→Y
X-R sin(Z)→K
Y-R (1-cos(Z))→P
(R+P) tan(I/2)+K→T
Y/sin(Z)→S
Pause
A-iPart(0.01T)→U
B-100 fPart(0.01T)→V
Disp “TS:”,U,”+”,V
Pause
Disp “SC:”,U+iPart(0.01L),”+”,V+100 fPart(0.01L)
Pause
Disp “ST:”,S

Casio fx-5800P Program HWYSPIRAL

Deg
“HIGHWAY SPIRAL”
“PI X: “?→A
“PI Y: “?→B
“INT-ANGLE: “?→I
“LENGTH: “?→L
“DEGREE: “?→D
L*D÷200→Z
100÷(D*π÷180)→R
Z*π÷180→Q
L*(1-Q²÷10)→X
L*(Q÷3-Q^(3)÷43)→Y
X-R sin(Z)→K
Y-R (1-cos(Z))→P
(R+P) tan(I÷2)+K→T
Y÷sin(Z)→S
“ANGLE TS-SC:”
Z⊿
R⊿
A-Int(0.01T)→U
B-100 Frac(0.01T)→V
“TS X:”
U⊿
“TS Y:”
V⊿
“SC X:”
U+Int(0.01L)⊿
“SC Y:”
V+100 Frac(0.01L)⊿
“ST:”
S

Example

A highway with a transition spiral is at station 50 + 64.84, with the intersection angle 50°, and the degree of the curve at 6°.  The length of the curve is said to be 360 ft.

Inputs:

PI X = 50
PI Y = 64.84
INT-ANGLE = 50
LENGTH = 360
DEGREE = 6

Outputs:

ANGLE TS-SC = 10.8