Method and system for rendering image on conical surface

1. An automated process for rendering an image on a conical surface, the automated process comprising:

by applying a transformation formula, transforming a two-dimensional (2D) planar representation of the image to a 2D planar representation of the image configured for a conical shape;

applying the transformed 2D planar representation of the image onto a planar template; and

manufacturing a conically-shaped object from the planar template on which the transformed 2D planar representation of the image has been applied,

wherein the transformation formula is represented as follows:

Xb =cos(radians(90−(θ*(( Xa*k )−(( Px*k )/2))/(( Px*k )/2))))*((( Py*k )−( Ya*k ))+( w−h ))

Yb=h ((( Py*k )−( Ya*k ))−(((( Py*k )−( Ya*k ))+( w−h ))−(cos((radians((θ*(( Xa*k )(( Px*k )/2))/(( Px*k )/2)))))*((( Py*k )−( Ya*k ))+( w−h )))))

where:

k (%): ratio correction for the xy grid post warping, k value is a ratio of the y-axis height of the 2D planar representation of the image (H1) to a y-axis height of the transformed 2D planar representation of the image configured for the conical shape (H2), which is based on the size of the specific conical shape concerned

θ (degrees): apex angle of the conical shape

Px: number of pixels on horizontal axis

Py: number of pixels on vertical axis

Xa: position of point on x-axis of rectangular grid

Ya: position of point on y-axis of rectangular grid

Xb: adjusted position on x-axis of warped grid

Yb: adjusted position on y-axis of warped grid

w: width of grid

h: height of grid

H1: y-axis height of the 2D planar representation of the image

H2: y-axis height of the transformed 2D planar representation of the image configured for the conical shape

k: H1/H2.

2. The automated process of claim 1 , comprising operating a processor to:

receive the 2D planar representation of the image;

determine dimensions of the conical shape; and

apply the transformation formula to the 2D planar representation according to the dimensions of the conical shape to obtain the 2D planar representation of the image configured for the conical shape.

3. The automated process of claim 2 , wherein the receiving the 2D planar representation of the image comprises receiving the planar template comprising the image.

4. The automated process of claim 3 , wherein the planar template comprises a rectangular template comprising the image.

5. The automated process of claim 1 , wherein the applying the transformed 2D planar representation of the image onto the planar template comprises printing the transformed 2D planar representation of the image onto the planar template.

6. The automated process of claim 1 , wherein the manufacturing the conically-shaped object from the planar template comprises forming the planar template into the required shape for the conically-shaped object.

7. The automated process of claim 1 , wherein the conically-shaped object is a paper cup.

8. The automated process of claim 1 , wherein the manufacturing the conically-shaped object from the printed planar template comprises using a conical object forming apparatus to produce the conically-shaped object.

9. An automated system for rendering an image on a conical surface, the automated system comprising:

a computing device comprising at least one processor for applying a transformation formula and resultantly transforming a two-dimensional (2D) planar representation of the image to a 2D planar representation of the image configured for a conical shape;

an image printing apparatus for printing the transformed 2D planar representation of the image onto a planar template; and

a conical object forming apparatus for manufacturing a conically-shaped object from the planar template on which the transformed 2D planar representation of the image has been applied,

wherein the transformation formula is represented as follows:

Xb =cos(radians(90−(θ*(( Xa*k )−(( Px*k )/2))/(( Px*k )/2))))*((( Py*k )−( Ya*k ))+( w−h ))

Yb=h ((( Py*k )−( Ya*k ))−(((( Py*k )−( Ya*k ))+( w−h ))−(cos((radians((θ*(( Xa*k )(( Px*k )/2))/(( Px*k )/2)))))*((( Py*k )−( Ya*k ))+( w−h )))))

where:

k (%): ratio correction for the xy grid post warping, k value is a ratio of the y-axis height of the 2D planar representation of the image (H1) to a y-axis height of the transformed 2D planar representation of the image configured for the conical shape (H2), which is based on the size of the specific conical shape concerned

θ (degrees): apex angle of the conical shape

Px: number of pixels on horizontal axis

Py: number of pixels on vertical axis

Xa: position of point on x-axis of rectangular grid

Ya: position of point on y-axis of rectangular grid

Xb: adjusted position on x-axis of warped grid

Yb: adjusted position on y-axis of warped grid

w: width of grid

h: height of grid

H1: y-axis height of the 2D planar representation of the image

H2: y-axis height of the transformed 2D planar representation of the image configured for the conical shape

k: H1/H2.

10. The automated system of claim 9 , wherein the at least one processor is configured to:

receive the 2D planar representation of the image;

determine dimensions of the conical shape; and

apply the transformation formula to the 2D planar representation according to the dimensions of the conical shape to obtain the 2D planar representation of the image configured for the conical shape.

11. The automated system of claim 10 , wherein the receiving the 2D planar representation of the image comprises receiving the planar template comprising the image.

12. The automated system of claim 11 , wherein the planar template comprises a rectangular template comprising the image.

13. The automated system of claim 10 , wherein the conical object forming apparatus is configured to cut the planar template into the required shape for the conically-shaped object.

14. The automated system of claim 10 , wherein the conical object forming apparatus comprises a paper cup forming apparatus.