I understand the dynamics change, but your post seems to imply that there is a hard divide. Did you overstate your belief or do you truly believe you should treat a 16 year old daughter like any other women not your wife.
Jim {If you do believe this, I truly feel sorry for you and your daughters}
So when my daughter grows breasts, she becomes a stranger?
Seriously, my daughter is fifteen, and we are very affectionate. We hug and kiss all the time, we cuddle on the sofa together when we watch tv, and she’d sit on my lap in a heartbeat, especially if it were the only seat available. Is that icky? Is it less icky because we’re both female? What if one of us were gay? Is that still okay?
Why is it so difficult to understand that not all physical contact is sexual?
Obviously you don’t treat your daughters exactly like your aunt. Nothing wrong with telling your daughters that they are your favorite person in the world. You can and should have a special intimate relationship with your daughters. It’s only in physical contact that you need to make adjustments as they mature. Obviously you stop bathing them at a certain age, you stop helping them dress at a certain age. There are certain things that are appropriate with a younger child that are inappropriate with an older child. I happen to think lap-sitting is one of those things.
Now, my aunt sitting on my lap–that’s creepy.
Especially because she’s dead.
So what are these ages? Is it a matter of developement? Or will you know it when you see it?
I only ask because you appear to be an expert on this. 
Jim {Please forgive the teasing}
To respect their privacy.
The situations I’m talking about are when I pass the bathroom after they shower and the door is open a crack.
Or as they are heading from the shower to their bedroom in a towel.
Or if I went to put something in their room and didn’t realize they were in there changing.
That kind of thing.
I don’t think of our family as excessively prudish, but we don’t generally walk around the house naked or in our underwear.
Of course, if we happen to see each other naked or in their underwear, we don’t make a big deal out of it.
It’s simpler than that; why should you have to justify the normal non-sexual interaction you have with your family, to people who for some reason find it ‘oogy’ or ‘icky’?
I think you’ll know when you see it. If you start to feel uneasy about your daughter in your lap, then it probably is time to stop. I’m no Doctor Spock hear, I’ve just raised a few kids. The Humble Opinion was asked for and given.
How about this for creepy- (my wife’s ex-hubby actually did this with his new wife and son) Would you let a teenaged son share the bed with you and your spouse? I think it shows we all draw the line a bit differently. Where and when you draw the lines are up to you.
Just out of interest, has anyone seen the movie Meet Joe Black? Did you find the portrayals of (entirely non-sexual) affection* between William Parrish (Anthony Hopkins) and his daughter Susan (Claire Forlani) to be ‘oogy’, ‘creepy’ or ‘icky’?
*To include hugging, kissing on the lips, cuddling on the sofa, holding hands…
But what if there’s not space? Stuff like this happens all the time in India - grandson sleeps with grandfather because there’s no more beds.
or in 19th century US–all those stories about Lincoln being gay because he shared a bed with a guy, a common non-sexual practice of his time.
I haven’t read all the posts, but I can’t even imagine it being creepy. I sat on my dad’s and grandfather’s laps. My daughter would probably still be sitting on her great-grandpa’s lap if he was still alive. We’re a close, loving, touchy family. Nothing dirty or perverted about that.
There is a HUGE difference between sitting on dad’s lap and sleeping with him and his new squeeze.
Let’s compare it to my 17 year old daughter laying under the covers on my bed with me and my SO while we watch a movie. It might sound creepy to you, but it’s perfectly innocent.
By the way, my seventeen year old sits on my lap all the time. Is it any less creepy with me being the mother?
Are you my dad? 
Like WhyNot said, sitting in someone’s lap is not always (or even usually) a sexual thing. There’s a picture of me perched on my dad’s knee at my wedding. It’s a sweet picture, and will probably end up framed somewhere in my parents’ house.
After which it will serve as “Exhibit A” in your dad’s child-abuse trial.
Our daughter-in-law is in her early 20s, and at a family wedding shower she sat on her father’s lap and draped an arm over his shoulder to talk to him for a few minutes. It was a lot less awkward than it would have been for her to bend over to talk to him, thus mooning the rest of us. I have to think that if she’s sat on that lap while learning to tie her shoes, she can certainly sit there to have a discrete discussion with the old man. I think the gesture strongly suggests, “Daddy, this is your little girl, and I need you to be the big strong hero you’ve always been.” It is entirely different from when she sits on her husband’s lap – her body language is different, and there’s definitely the attitude of something promised that was completely lacking when she sat on her Pop’s lap.
Nobody’s talking about snuggling and petting, both of which are overtly sexual activities and either foreplay to or substitution for sexual intercourse. My family did and still does hug, and when we get back together after not seeing each other for months or years, they are tight, manly clinches. We don’t pet each other. And yes, my mother has sat on my lap, when I was lecturing her about skipping an appointment with her cardiologist. In a gesture filled with irony and good humor, she sat on my lap, wagged her finger in my face and proceeded to tell me off – good-naturedly, of course.
Obviously, the action has to be taken in context. I’m guessing that people who refuse to consider circumstance or context are moral absolutists who are quick to judge other people’s intentions based on their own issues.
After reading most of this thread, I gotta’ say, there is a very large group of people out there who have serious issues with family members exchanging physical displays of affection. I think they equate PDAs with sex, and to me that means they’re the ones with dirty minds, not the rest of us.
First of all, I don’t think I’ve read anyone say that no one over the age of 11.5 should ever sit on any family member for any reason whatsoever.
And speaking of judging, why do you have to label everyone that doesn’t agree with you as having “dirty minds”? As other people have already pointed out: Some think it’s okay to bounce a baby on your knee, but not a teenager. Some people think it’s okay to bathe a toddler, and not bathe a teenager. Some think it’s okay to raspberry a toddler’s tummy, but not as okay to raspberry your mom. Even though there’s nothing sexual about blowing raspberries! Just because some people draw the line in a different place than you do, doesn’t mean they’re stinted, dirty, prudish or have serious issues. Sheesh.
I really hope you’re kidding. If not, you’re a pretty sick person.
I don’t see any problem with parents being affectionate with their children, even to this extent.
I would love to be able to sit on my dad’s lap, or even give him a tight hug. Ever since his cancer broke his back, I can barely touch him. When he was able to put his arms around me when he had healed well enough to use a cane, I had to go in the bathroom and cry so he wouldn’t see me, because I was so happy to be able to hug him again. So, yeah, I’m a little jeolous of people that could conceivably still have normal contact with their dads, and I know that good fathers have absolutely no sexual feelings toward their daughters. And if a daughter didn’t trust her father to be truly platonic about contact, I doubt that most would get so close anyway.
Sorry about getting so glurgy, but that’s honestly how I feel about it. If you have a healthy, loving family, physical contact isn’t just normal, it’s necessary.