Engineering ToolBox - enabled for use with the amazing FREE SketchUp 3D drawing and modeling program.

## Bezier Curve - in Equal Segments

A 4 points Bezier Curve split in a custum no. of segments with equal lengths. This tool can be useful when constructing walkable U- and L-shaped stairs

A four points Bezier curve can be expressed with the equation:

P(i) = P0 (1-i)3 + P1 3 i (1-i)2 + P2 3 i2 (1-t) + P3 i3

where the interval - i - runs from 0 to 1

Add coordinates of control points, no. of intervals and no. of segments - - and insert a segmented Bezier curve to your model. In addition to line segments small lines are added to indicate the angle between the segments. These lines can used to construct steps in a stair as indicated in the example bellow.

Note! - due to the nature of the Bezier equation the length of the segments will not be exactly equal. In general: smaller intervals increases accuracy.

This Bezier curve with equal segments don't work on Windows computers. The normal Bezier curve may be helpful.

 red green blue P0 - start point P0SwitchP1 P1 - control point P2 - control point P2SwitchP3 P3 - end point Insert Bezier Curve from the Points above! Control Points from two Selected Lines? Check here! i - interval (0 - 1) (in the Bez. eq.) no. of segments (split the Bez. curve) m dm cm mm in ft Calculated Segments

 Length m dm cm mm in ft Color no color aqua black blue fuchsia green gray lime maroon navy olive purple red silver teal white yellow Offset m dm cm mm in ft

### Example - Construction of a L-shaped stair

Note that there are limits to the steps rise and run for a stair to be walkable. Check other stairs for more information. In addition - for a step to be walkable - the steps rise and run should be more or less constant in the "natural path" of the stair.

The "natural path" differ with persons and for persons over time - but a good aproximation to the "natural path" is the Bezier curve. This tool inserts a Bezier curve in what you define as the "natural path" - and split the curve in aprox. equal length segments. These segments can be used for the further construction of the stair.

#### Procedure

• draw the footprint of the staircase
• draw the control lines for the Bezier curve
• measure the coordinates of the control lines and add them to the inputs above - use the "Get control points.." feature above, or from the dialog box, or use the "Tools > Text" feature in Sketchup - as used in the example above.
• adjust the interval parameter if required (lower value - in general higher accuracy)
• set the numbers of steps
• insert the Bezier curve!
• repeat - change the control points until you find the "natural path" you believe is the best for the stair
• extend the small lines between the segments until they touch the walls of the staircase
• use the push/pull tool to "lift" the areas representing the steps to their exact levels. Tip! - insert a stair with the right height and the right number of steps as a reference for the push/pull tool
• customize the stair and the steps - modify step noses, details ...

## Tools - accessed from within SketchUp

NOTE!! - the "insert!" and the Tools features works only from within the engineeringtoolbox plug-in inside of SketchUp. Install the plugin/extension as described in the right column - restart Sketchup and open this tool from the Sketchup Plugins/Extensions menu.

undo! - redo!

### Color

change color! - of the selected object(s) - may be slow for some more complicated objects (save your model first!)

### View

iso! - top! - front! - right! - left! - back! - bottom!

### Make Component

make! - makes a component of the selected object. Note that all objects from the Engineering Toolbox are inserted as individual groups. You should transfer a group to a component sharing the same definition when there is more than one instance of the same object. Tips! Rename the object to its final name before making it a component (in outliner or with right click and entity info).

### Rotate Object(s)

degrees around red, green or blue axis! (select object(s) first). Very handy for objects with curved surfaces like pipes, elbows ..

### Move or Copy Object(s)

move/copy! no. copies red green blue (offset from current position
- move or copy (multiple) selected objects. Make advanced geometric constructions with multiple selections, multiple copies and the offset.
Tip! - Use "Make Component" to save space and reduce computer load when working with multiple copies.

### The Knife

- inserts a "Knife" to cut in one or more objects.
insert the Knife! size of the Knife
cut! - before cut: customize the inserted Knife! The Knife is a group you can rotate, change the size of, and even add or remove objects within (example - add functional curves for advanced cuts). You can also make your own Knife by naming a group "theKnife". After insert and customizing the knife - positioning it to the object(s) where to cut - select the object(s) (must be groups) - and click cut! Afterwards - select the object(s) and remove the parts of the object(s) that you don't want.

### Move an Object to another Objects Position

move! - select two objects - then click "move" and one of the objects moves to the position and angle of the other object. Handy when moving between axis systems.

### Change Insertion Point

You can change the position and angle of an inserting object by pre-selecting an existing object before Insert! The inserted object will inherit the position of the selected object. Handy when working with non-default axis.

### Insert Line

red green blue - Start position
red green blue - End position
insert! - use this feature to add a reference line

### Insert Horizontal Plane

insert! red (size) green (size) blue (vertical position)
- use this feature to add a horizontal reference plane

### Insert Vertical Plane

insert! red (horizontal position) green (size) blue (vertical size)
insert! red (size) green (horizontal position) blue (vertical size)
- use this feature to add vertical a reference plane

### Insert Box

insert! red (width) green (depth) blue (height)
- use this feature to add a box to the model

### Insert external Component

insert! name url
- use this feature to add an external component from the internet by adding the components url